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''measure of a set''
 
''measure of a set''
  
 
A notion that generalizes those of the length of segments, the area of figures and the volume of bodies, and that corresponds intuitively to the mass of a set for some mass distribution throughout the space. The notion of the measure of a set arose in the theory of functions of a real variable in connection with the study and improvement of the notion of an [[Integral|integral]].
 
A notion that generalizes those of the length of segments, the area of figures and the volume of bodies, and that corresponds intuitively to the mass of a set for some mass distribution throughout the space. The notion of the measure of a set arose in the theory of functions of a real variable in connection with the study and improvement of the notion of an [[Integral|integral]].
 
 
  
 
==Definition and general properties.==
 
==Definition and general properties.==
Line 80: Line 77:
  
 
A pair $ (X, \mathcal{S}) $, where $ X $ is a set and $ \mathcal{S} $ is a $ \sigma $-ring of subsets of $ X $ such that $ \bigcup_{E \in \mathcal{S}} E = X$, is called a [[Measurable space|measurable space]]. A triple $ (X, \mathcal{S}, \mu) $, where $ (X, \mathcal{S}) $ is a measurable space and $ \mu $ is a measure on $ \mathcal{S} $, is called a [[Measure space|measure space]]. A space with a totally-finite measure $ \mu $ normalized by the condition $ \mu(X) = 1 $ is called a [[Probability space|probability space]]. In abstract measure theory, where the basic notions are a measurable space $ (X, \mathcal{S}) $ or a measure space $ (X, \mathcal{S}, \mu) $, the elements of $ \mathcal{S} $ are also referred to as measurable sets (cf. also [[Measurable set|Measurable set]]).
 
A pair $ (X, \mathcal{S}) $, where $ X $ is a set and $ \mathcal{S} $ is a $ \sigma $-ring of subsets of $ X $ such that $ \bigcup_{E \in \mathcal{S}} E = X$, is called a [[Measurable space|measurable space]]. A triple $ (X, \mathcal{S}, \mu) $, where $ (X, \mathcal{S}) $ is a measurable space and $ \mu $ is a measure on $ \mathcal{S} $, is called a [[Measure space|measure space]]. A space with a totally-finite measure $ \mu $ normalized by the condition $ \mu(X) = 1 $ is called a [[Probability space|probability space]]. In abstract measure theory, where the basic notions are a measurable space $ (X, \mathcal{S}) $ or a measure space $ (X, \mathcal{S}, \mu) $, the elements of $ \mathcal{S} $ are also referred to as measurable sets (cf. also [[Measurable set|Measurable set]]).
 
 
 
 
  
 
==Properties of measure spaces.==
 
==Properties of measure spaces.==
Line 109: Line 102:
  
 
\[\mu T^{-1}(E) = \mu(T^{-1}E).\]
 
\[\mu T^{-1}(E) = \mu(T^{-1}E).\]
 
  
 
Let $(X, \mathcal{S}, \mu)$ be a measure space and let $X_1 \subset X$. Define $\mu_{X_1}$ on the sets $E$ from the $\sigma$-ring $ \mathcal{S} \cap X_1 = \{ E \cap X_1: E \in \mathcal{S}_1\}$ by
 
Let $(X, \mathcal{S}, \mu)$ be a measure space and let $X_1 \subset X$. Define $\mu_{X_1}$ on the sets $E$ from the $\sigma$-ring $ \mathcal{S} \cap X_1 = \{ E \cap X_1: E \in \mathcal{S}_1\}$ by
Line 123: Line 115:
  
 
==Extension of measures.==
 
==Extension of measures.==
A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240203.png" /> is an extension of a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240204.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240205.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240206.png" /> in the class of finitely-additive measures (see above). Every measure defined on a semi-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240207.png" /> admits a unique extension to a measure on the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240208.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240209.png" /> (the extension is realized in the same way as in the case of finitely-additive measures). Further, every measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240210.png" /> defined on a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240211.png" /> can be extended to a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240212.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240213.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240214.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240215.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240216.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240217.png" />-finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240218.png" /> is unique and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240219.png" />-finite. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240220.png" /> on any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240221.png" /> can be given by the formula
+
A measure $  \mu _{2} $
 +
is an extension of a measure $  \mu _{1} $
 +
if $  \mu _{2} $
 +
is an extension of $  \mu _{1} $
 +
in the class of finitely-additive measures (see above). Every measure defined on a semi-ring $  {\mathcal P} $
 +
admits a unique extension to a measure on the ring $  {\mathcal R}( {\mathcal P}) $
 +
generated by $  {\mathcal P} $(
 +
the extension is realized in the same way as in the case of finitely-additive measures). Further, every measure $  \mu $
 +
defined on a ring $  {\mathcal R} $
 +
can be extended to a measure $  \mu^ \prime  $
 +
on the $  \sigma $-
 +
ring $  {\mathcal S} ( {\mathcal R}) $
 +
generated by $  {\mathcal R} $;  
 +
if $  \mu $
 +
is $  \sigma $-
 +
finite, then $  \mu^ \prime  $
 +
is unique and $  \sigma $-
 +
finite. The value of $  \mu^ \prime  $
 +
on any set $  E \in {\mathcal S} ( {\mathcal R}) $
 +
can be given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240222.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{*}
 +
\mu^ \prime  (E) \  = \  \mathop{\rm inf}\nolimits \left \{ {\sum _ { i=1 } ^ \infty
 +
\mu (E _{i} )} : {E _{i} \in {\mathcal R} ,\  i = 1,\  2 \dots \  E \subset  \cup _ { i=1 } ^ \infty
 +
E _ i} \right \}
 +
.
 +
$$
  
A class of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240223.png" /> is called hereditary if it contains, together with any set in the class, all its subsets. An [[Outer measure|outer measure]] is a set function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240224.png" />, defined on a hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240226.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240227.png" /> (i.e. a class of sets which is simultaneously hereditary and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240228.png" />-ring), which has the following properties:
+
A class of subsets of $  X $
 +
is called hereditary if it contains, together with any set in the class, all its subsets. An [[Outer measure|outer measure]] is a set function m^ \star  $,  
 +
defined on a hereditary $  \sigma $-
 +
ring $  {\mathcal H} $(
 +
i.e. a class of sets which is simultaneously hereditary and a $  \sigma $-
 +
ring), which has the following properties:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240229.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240230.png" />;
+
1) 0 \leq  m^ \star  (E) \leq  \infty $,  
 +
$  m^ \star  (\emptyset) = 0 $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240231.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240232.png" />;
+
2) $  E \subset  F $
 +
implies  $  m^ \star  (E) \leq  m^ \star  (F \  ) $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240233.png" />.
+
3) $  m^ \star  ( \cup _ i=1^ \infty  E _{i} ) \leq  \sum _ i=1^ \infty  m^ \star  (E _{i} ) $.
  
Given a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240234.png" /> on the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240235.png" /> one can construct an outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240236.png" /> on the hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240237.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240238.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240239.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240240.png" /> consists of all sets that can be covered by a countable union of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240241.png" />) by means of the formula
+
Given a measure $  \mu $
 +
on the ring $  {\mathcal R} $
 +
one can construct an outer measure $  \mu^ \star  $
 +
on the hereditary $  \sigma $-
 +
ring $  {\mathcal H} ( {\mathcal R}) $
 +
generated by $  {\mathcal R} $(
 +
$  {\mathcal H}( {\mathcal R}) $
 +
consists of all sets that can be covered by a countable union of elements of $  {\mathcal R} $)  
 +
by means of the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240242.png" /></td> </tr></table>
+
$$
 +
\mu^ \star  (E) \  = \  \mathop{\rm inf}\nolimits \left \{ {\sum _ { i=1 } ^ \infty \mu (E _{i} )} : {E _{i} \in {\mathcal R} ,\
 +
i = 1,\  2 \dots \  E \subset  \cup _ { i=1 } ^ \infty E _ i} \right \} .
 +
$$
  
The outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240243.png" /> is called the outer measure induced by the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240244.png" />.
+
The outer measure $  \mu^ \star  $
 +
is called the outer measure induced by the measure $  \mu $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240245.png" /> be an outer measure on a hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240246.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240247.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240248.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240249.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240251.png" />-measurable if
+
Let m^ \star  $
 +
be an outer measure on a hereditary $  \sigma $-
 +
ring $  {\mathcal H} $
 +
of subsets of $  X $.  
 +
A set $  E \in {\mathcal H} $
 +
is called m^ \star  $-
 +
measurable if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240252.png" /></td> </tr></table>
+
$$
 +
m^ \star  (A) \  = \  m^ \star  (A \cap E) + m^ \star  (A
 +
\cap (X \setminus  E))
 +
$$
  
for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240253.png" />. The collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240254.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240255.png" />-measurable sets is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240256.png" />-ring which contains all sets of null outer measure. The set function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240257.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240258.png" /> defined by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240259.png" /> is a complete measure and is called the measure induced by the outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240260.png" />.
+
for every $  A \in {\mathcal H} $.  
 +
The collection $  \overline{ {\mathcal S} }\; $
 +
of m^ \star  $-
 +
measurable sets is a $  \sigma $-
 +
ring which contains all sets of null outer measure. The set function $  \overline{m}\; $
 +
on $  \overline{ {\mathcal S} }\; $
 +
defined by the equality $  \overline{m}\; (E) = m^ \star  (E) $
 +
is a complete measure and is called the measure induced by the outer measure m^ \star  $.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240261.png" /> is a measure on a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240262.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240263.png" /> is the outer measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240264.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240265.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240266.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240267.png" /> denote the collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240268.png" />-measurable sets and the measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240269.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240270.png" />, respectively. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240271.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240272.png" />, and since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240273.png" /> it follows that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240274.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240275.png" /> given by formula (*) is also a measure extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240276.png" />. If the original measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240277.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240278.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240279.png" />-finite, then the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240280.png" /> is the completion of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240281.png" /> (see (*)). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240282.png" /> is given on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240283.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240284.png" />, then the induced outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240285.png" /> on the hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240286.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240287.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240288.png" /> is given by the formula
+
Suppose that $  \mu $
 +
is a measure on a ring $  {\mathcal R} $
 +
and that $  \mu^ \star  $
 +
is the outer measure on $  {\mathcal H} ( {\mathcal R}) $
 +
induced by $  \mu $.  
 +
Let $  \overline{ {\mathcal S} }\; $
 +
and $  \overline \mu \; $
 +
denote the collection of $  \mu^ \star  $-
 +
measurable sets and the measure on $  \overline{ {\mathcal S} }\; $
 +
induced by $  \mu^ \star  $,  
 +
respectively. Then $  \overline \mu \; $
 +
is an extension of $  \mu $,  
 +
and since $  {\mathcal S} ( {\mathcal R}) \subset  \overline{ {\mathcal S} }\; $
 +
it follows that the function $  \mu^ \prime  $
 +
on $  {\mathcal S}( {\mathcal R}) $
 +
given by formula (*) is also a measure extending $  \mu $.  
 +
If the original measure $  \mu $
 +
on $  {\mathcal R} $
 +
is $  \sigma $-
 +
finite, then the space $  (X,\  \overline{ {\mathcal S} }\; ,\  \overline \mu \; ) $
 +
is the completion of the space $  (X,\  {\mathcal S}( {\mathcal R}) ,\  \mu^ \prime  ) $(
 +
see (*)). If $  \mu $
 +
is given on the $  \sigma $-
 +
ring $  {\mathcal S} $,  
 +
then the induced outer measure $  \mu^ \star  $
 +
on the hereditary $  \sigma $-
 +
ring $  {\mathcal H}( {\mathcal S}) $
 +
generated by $  {\mathcal S} $
 +
is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240289.png" /></td> </tr></table>
+
$$
 +
\mu^ \star  (E) \  = \  \mathop{\rm inf}\nolimits \{ {\mu (F \  )} : {E \subset  F,\  F
 +
\in {\mathcal S}} \}
 +
.
 +
$$
  
Alongside with the outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240290.png" />, one defines the inner measure induced by the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240291.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240292.png" />. It is defined as
+
Alongside with the outer measure $  \mu^ \star  $,  
 +
one defines the inner measure induced by the measure $  \mu $
 +
on $  {\mathcal S} $.  
 +
It is defined as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240293.png" /></td> </tr></table>
+
$$
 +
\mu _ \star  (E) \  = \  \sup \{ {\mu (F \  )} : {E \supset F,\  F
 +
\in {\mathcal S}} \}
 +
,\ \  E \in
 +
{\mathcal H}( {\mathcal S}).
 +
$$
  
For every set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240294.png" /> a measurable kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240295.png" /> and a measurable envelope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240296.png" /> are defined as elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240297.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240298.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240299.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240300.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240302.png" />. A measurable kernel exists always, while a measurable envelope exists whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240303.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240304.png" />-finite outer measure; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240306.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240307.png" /> be a measure on a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240308.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240309.png" /> be its extension to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240310.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240311.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240312.png" />. The inner measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240313.png" /> on the subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240314.png" /> of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240315.png" />-measure can be expressed in terms of the outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240316.png" /> (and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240317.png" />):
+
For every set $  E \in {\mathcal H}( {\mathcal S}) $
 +
a measurable kernel $  E^ \prime  $
 +
and a measurable envelope $  E^{\prime\prime} $
 +
are defined as elements of $  {\mathcal S} $
 +
such that $  E^ \prime  \subset  E \subset  E^{\prime\prime} $
 +
and $  \mu (F ^ {\  \prime} ) = \mu (F ^ {\  \prime\prime} ) = 0 $
 +
for all $  F ^ {\  \prime} ,\  F ^ {\  \prime\prime} \in {\mathcal S} $
 +
such that $  F ^ {\  \prime} \subset  E\setminus E^ \prime  $,  
 +
$  F ^ {\  \prime\prime} \subset  E^{\prime\prime} \setminus E $.  
 +
A measurable kernel exists always, while a measurable envelope exists whenever $  E $
 +
has $  \sigma $-
 +
finite outer measure; moreover, $  \mu _ \star  (E) = \mu (E^ \prime  ) $
 +
and $  \mu^ \star  (E) = \mu (E^{\prime\prime} ) $.  
 +
Let $  \mu $
 +
be a measure on a ring $  {\mathcal R} $
 +
and let $  \mu^ \prime  $
 +
be its extension to the $  \sigma $-
 +
ring $  {\mathcal S}( {\mathcal R}) $
 +
generated by $  {\mathcal R} $.  
 +
The inner measure $  \mu _ \star^ \prime  $
 +
on the subsets $  E $
 +
of finite $  \mu $-
 +
measure can be expressed in terms of the outer measure $  \mu^ \star  $(
 +
and hence $  \mu $):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240318.png" /></td> </tr></table>
+
$$
 +
\mu _ \star^ \prime  (A) \  = \  \mu (E) - \mu^ \star  (E \setminus A),\ \
 +
A \subset  E.
 +
$$
  
Furthermore, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240319.png" /> belonging to the hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240320.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240321.png" /> with finite outer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240322.png" />-measure is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240323.png" />-measurable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240324.png" />. In case the original measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240325.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240326.png" /> is totally finite, one has the following necessary and sufficient condition for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240327.png" />-measurability of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240328.png" />:
+
Furthermore, a set $  F $
 +
belonging to the hereditary $  \sigma $-
 +
ring $  {\mathcal H}( {\mathcal R}) $
 +
with finite outer $  \mu^ \star  $-
 +
measure is $  \mu^ \star  $-
 +
measurable if and only if $  \mu^ \star  (F \  ) = \mu _ \star^ \prime  (F \  ) $.  
 +
In case the original measure $  \mu $
 +
on $  {\mathcal R} $
 +
is totally finite, one has the following necessary and sufficient condition for the $  \mu^ \star  $-
 +
measurability of a set $  E \subset  X $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240329.png" /></td> </tr></table>
+
$$
 +
\mu (X) \  = \  \mu^ \star  (E) + \mu^ \star  (X\setminus E).
 +
$$
  
For totally-finite measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240330.png" /> this condition is frequently taken as the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240331.png" />-measurability of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240332.png" />.
+
For totally-finite measures on $  {\mathcal R} $
 +
this condition is frequently taken as the definition of $  \mu^ \star  $-
 +
measurability of the set $  E $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240333.png" /> is a space with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240334.png" />-finite measure and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240335.png" /> is a finite collection of elements of the hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240336.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240337.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240338.png" />, then on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240339.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240340.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240341.png" /> and the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240342.png" /> one can define a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240343.png" /> which agrees with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240344.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240345.png" />.
+
If $  (X,\  {\mathcal S} ,\  \mu ) $
 +
is a space with a $  \sigma $-
 +
finite measure and $  X _{1} \dots X _{n} $
 +
is a finite collection of elements of the hereditary $  \sigma $-
 +
ring $  {\mathcal H}( {\mathcal S}) $
 +
generated by $  {\mathcal S} $,  
 +
then on the $  \sigma $-
 +
ring $  \widetilde{ {\mathcal S} }  $
 +
generated by $  {\mathcal S} $
 +
and the sets $  X _{1} \dots X _{n} $
 +
one can define a measure $  \widetilde \mu  $
 +
which agrees with $  \mu $
 +
on $  {\mathcal S} $.
  
 
==Jordan, Lebesgue and Lebesgue–Stieltjes measures.==
 
==Jordan, Lebesgue and Lebesgue–Stieltjes measures.==
An example of an extension of a measure is provided by the Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240346.png" />. The intervals of the form
+
An example of an extension of a measure is provided by the Lebesgue measure in $  \mathbf R^{k} $.  
 +
The intervals of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240347.png" /></td> </tr></table>
+
$$
 +
I \  = \  \{ {(x _{1} \dots x _{k} )} : {a _{i} \leq  x _{i} < b
 +
_{i} ,\
 +
i = 1 \dots k} \}
 +
$$
  
form a semi-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240348.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240349.png" />. For each such interval, let
+
form a semi-ring $  {\mathcal P} $
 +
in $  \mathbf R^{k} $.  
 +
For each such interval, let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240350.png" /></td> </tr></table>
+
$$
 +
\lambda (I) \  = \  \prod _ { i=1 } ^ k (b _{i} - a _{i} )
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240351.png" /> coincides with the volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240352.png" />). The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240353.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240354.png" />-finite and countably additive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240355.png" /> and admits a unique extension to a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240356.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240357.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240358.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240359.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240360.png" /> is identical with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240361.png" />-ring of Borel sets (cf. [[Borel set|Borel set]]) (or Borel-measurable sets) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240362.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240363.png" /> was first defined by E. Borel in 1898 (see [[Borel measure|Borel measure]]). The completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240364.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240365.png" /> (defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240366.png" />) is called the Lebesgue measure, and was introduced by H. Lebesgue in 1902 (see [[Lebesgue measure|Lebesgue measure]]). A set belonging to the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240367.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240368.png" /> is called Lebesgue measurable. A bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240369.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240370.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240371.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240372.png" /> is some interval containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240373.png" />; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240374.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240375.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240376.png" /> if and only if for some sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240377.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240378.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240379.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240380.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240381.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240382.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240383.png" />. The cardinality of the family of all Borel sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240384.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240385.png" /> (the cardinality of the continuum), whereas the cardinality of the family of all Lebesgue-measurable sets is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240386.png" />, so that the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240387.png" /> is strict, i.e. there exist Lebesgue-measurable sets that are not Borel measurable.
+
( $  \lambda (I) $
 +
coincides with the volume of $  I $).  
 +
The function $  \lambda $
 +
is $  \sigma $-
 +
finite and countably additive on $  {\mathcal P} $
 +
and admits a unique extension to a measure $  \lambda^ \prime  $
 +
on the $  \sigma $-
 +
ring $  {\mathcal S} $
 +
generated by $  {\mathcal P} $;  
 +
$  {\mathcal S} $
 +
is identical with the $  \sigma $-
 +
ring of Borel sets (cf. [[Borel set|Borel set]]) (or Borel-measurable sets) in $  \mathbf R^{k} $.  
 +
The measure $  \lambda^ \prime  $
 +
was first defined by E. Borel in 1898 (see [[Borel measure|Borel measure]]). The completion $  \overline \lambda \; $
 +
of $  \lambda^ \prime  $(
 +
defined on $  \overline{ {\mathcal S} }\; \  $)  
 +
is called the Lebesgue measure, and was introduced by H. Lebesgue in 1902 (see [[Lebesgue measure|Lebesgue measure]]). A set belonging to the domain $  \overline{ {\mathcal S} }\; $
 +
of $  \overline \lambda \; $
 +
is called Lebesgue measurable. A bounded set $  E \subset  \mathbf R^{k} $
 +
belongs to $  \overline{ {\mathcal S} }\; $
 +
if and only if $  \lambda (I) = \lambda^ \star  (E) + \lambda^ \star  (I\setminus E) $,  
 +
where $  I \in {\mathcal P} $
 +
is some interval containing $  E $;  
 +
in this case $  \overline \lambda \; (E) = \lambda^ \star  (E) $.  
 +
A set $  E \subset  \mathbf R^{k} $
 +
belongs to $  \overline{ {\mathcal S} }\; $
 +
if and only if for some sequence $  \{ r _{n} \} $,
 +
$  r _{n} > 0 $,  
 +
$  n = 1,\  2 \dots $
 +
such that $  r _{n} \rightarrow \infty $,  
 +
one has $  E \cap B _{ {r _ n}} \in \overline{ {\mathcal S} }\; $
 +
for all $  n $,  
 +
where $  B _{r} = \{ {x \in \mathbf R ^ k} : {\| x \| \leq  r} \} $.  
 +
The cardinality of the family of all Borel sets in $  \mathbf R^{k} $
 +
is $  \mathfrak c $(
 +
the cardinality of the continuum), whereas the cardinality of the family of all Lebesgue-measurable sets is $  2^{\mathfrak c} $,  
 +
so that the inclusion $  {\mathcal S} \subset  \overline{ {\mathcal S} }\; $
 +
is strict, i.e. there exist Lebesgue-measurable sets that are not Borel measurable.
  
The Lebesgue measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240388.png" /> is invariant under linear orthogonal transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240389.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240390.png" /> as well as under translations by elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240391.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240392.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240393.png" />.
+
The Lebesgue measure $  \overline \lambda \; $
 +
is invariant under linear orthogonal transformations $  A $
 +
of $  \mathbf R^{k} $
 +
as well as under translations by elements $  x \in \mathbf R^{k} $,  
 +
i.e. $  \overline \lambda \; ( A E + x) = \overline \lambda \; (E) $
 +
for all $  E \in {\mathcal S} $.
  
Using the [[Axiom of choice|axiom of choice]] one can show that there exist sets which are not Lebesgue measurable. On the straight line, for example, such a set can be obtained by picking one point in each coset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240394.png" /> of the additive subgroup of rational numbers (Vitali's example).
+
Using the [[Axiom of choice|axiom of choice]] one can show that there exist sets which are not Lebesgue measurable. On the straight line, for example, such a set can be obtained by picking one point in each coset in $  \mathbf R $
 +
of the additive subgroup of rational numbers (Vitali's example).
  
Historically the Borel and Lebesgue measures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240395.png" /> were preceded by the measure defined by C. Jordan in 1892 (see [[Jordan measure|Jordan measure]]). The idea of the definition of the Jordan measure is very close to that of the classic definition of area and volume, which goes back to ancient Greece. Thus, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240396.png" /> is called Jordan measurable if there exist two sets, representable as finite unions of disjoint rectangles, one contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240397.png" /> and the other containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240398.png" />, such that the difference of their volumes (defined in an obvious manner) is arbitrarily small. The Jordan measure of such a set is the infimum of the volumes of finite unions of rectangles covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240399.png" />. A Jordan-measurable set is also Lebesgue measurable, and its Jordan and Lebesgue measures are equal. The domain of the Jordan measure is merely a ring, and not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240400.png" />-ring, which restricts considerably its domain of applicability.
+
Historically the Borel and Lebesgue measures in $  \mathbf R^{k} $
 +
were preceded by the measure defined by C. Jordan in 1892 (see [[Jordan measure|Jordan measure]]). The idea of the definition of the Jordan measure is very close to that of the classic definition of area and volume, which goes back to ancient Greece. Thus, a set $  E \subset  \mathbf R^{k} $
 +
is called Jordan measurable if there exist two sets, representable as finite unions of disjoint rectangles, one contained in $  E $
 +
and the other containing $  E $,  
 +
such that the difference of their volumes (defined in an obvious manner) is arbitrarily small. The Jordan measure of such a set is the infimum of the volumes of finite unions of rectangles covering $  E $.  
 +
A Jordan-measurable set is also Lebesgue measurable, and its Jordan and Lebesgue measures are equal. The domain of the Jordan measure is merely a ring, and not a $  \sigma $-
 +
ring, which restricts considerably its domain of applicability.
  
The Lebesgue measure is a particular case of the more general Lebesgue–Stieltjes measure. The latter is defined by means of a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240401.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240402.png" /> with the properties:
+
The Lebesgue measure is a particular case of the more general Lebesgue–Stieltjes measure. The latter is defined by means of a real-valued function $  F $
 +
on $  \mathbf R^{k} $
 +
with the properties:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240403.png" />;
+
1) $  - \infty < F < \infty $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240404.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240406.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240407.png" /> is the difference operator with step <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240408.png" /> taken at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240409.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240410.png" />-th coordinate;
+
2) $  \Delta _{ {b _{1} - a _ 1}} \dots \Delta _{ {b _{k} - a _ k}} F(a _{1} \dots a _{k} ) \geq 0 $
 +
for $  a _{i} < b _{i} $,  
 +
$  i = 1 \dots k $,  
 +
where $  \Delta _{ {b _{i} - a _ i}} $
 +
is the difference operator with step $  b _{i} - a _{i} $
 +
taken at the point $  a _{i} $
 +
with respect to the $  i $-
 +
th coordinate;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240411.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240412.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240413.png" />.
+
3) $  F(a _{1} \dots a _{k} ) \uparrow F(b _{1} \dots b _{k} ) $
 +
as $  a _{i} \uparrow b _{i} $,  
 +
$  i = 1 \dots k $.
  
Given such a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240414.png" />, the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240415.png" /> of the interval
+
Given such a function $  F $,  
 +
the measure $  \mu _{F} $
 +
of the interval
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240416.png" /></td> </tr></table>
+
$$
 +
I \  = \  \{ {(x _{1} \dots x _{k} )} : {a _{i} \leq  x _{i} < b
 +
_{i} ,\  i = 1 \dots k} \}
 +
$$
  
 
is defined by the formula
 
is defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240417.png" /></td> </tr></table>
+
$$
 +
\mu _{F} (I) \  = \  \Delta _{ {b _{1} - a _ 1}} \dots \Delta
 +
_{ {b _{k} - a _ k}} F(a _{1} \dots a _{k} ).
 +
$$
  
It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240418.png" /> is countably additive on the semi-ring of all such intervals and that it admits an extension to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240419.png" />-algebra of Borel sets; the completion of this extension yields what is called the Lebesgue–Stieltjes measure corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240420.png" />. For the particular choice
+
It turns out that $  \mu _{F} $
 +
is countably additive on the semi-ring of all such intervals and that it admits an extension to the $  \sigma $-
 +
algebra of Borel sets; the completion of this extension yields what is called the Lebesgue–Stieltjes measure corresponding to $  F $.  
 +
For the particular choice
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240421.png" /></td> </tr></table>
+
$$
 +
F(x _{1} \dots x _{k} ) \  = \  x _{1} \dots x _{k}  $$
  
 
one obtains the Lebesgue measure.
 
one obtains the Lebesgue measure.
Line 208: Line 431:
 
==Measures in product spaces.==
 
==Measures in product spaces.==
 
{{Anchor|product}}
 
{{Anchor|product}}
By definition, the product of two measurable spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240422.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240423.png" /> is the measurable space consisting of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240424.png" /> (the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240425.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240426.png" />) and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240427.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240428.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240429.png" /> (the product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240430.png" />-rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240431.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240432.png" />) generated by the semi-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240433.png" /> of sets of the form
+
By definition, the product of two measurable spaces $  (X _{1} ,\  {\mathcal S} _{1} ) $,
 +
$  (X _{2} ,\  {\mathcal S} _{2} ) $
 +
is the measurable space consisting of the set $  X _{1} \times X _{2} = \{ {(x _{1} ,\  x _{2} )} : {x _{1} \in X _{1} ,\  x _{2} \in X _ 2} \} $(
 +
the product of $  X _{1} $
 +
and $  X _{2} $)  
 +
and the $  \sigma $-
 +
ring $  {\mathcal S} _{1} \times {\mathcal S} _{2} $
 +
of subsets of $  X $(
 +
the product of the $  \sigma $-
 +
rings $  {\mathcal S} _{1} $
 +
and $  {\mathcal S} _{2} $)  
 +
generated by the semi-ring $  {\mathcal P} $
 +
of sets of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240434.png" /></td> </tr></table>
+
$$
 +
E _{1} \times E _{2} \  = \
 +
\{ {(x _{1} ,\  x _{2} )} : {x _{1} \in E _{1} ,\
 +
x _{2} \in E _ 2} \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240435.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240436.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240437.png" /> are measure spaces, the formula
+
where $  E _{1} ,\  E _{2} \in {\mathcal S} $.  
 +
If $  (X _{1} ,\  {\mathcal S} _{1} ,\  \mu _{1} ) $
 +
and $  (X _{2} ,\  {\mathcal S} _{2} ,\  \mu _{s} ) $
 +
are measure spaces, the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240438.png" /></td> </tr></table>
+
$$
 +
\mu (E _{1} \times E _{2} ) \  = \
 +
\mu _{1} (E _{1} ) \mu _{2} (E _{2} ),\ \
 +
E _{1} \in {\mathcal S} _{1} ,\ \
 +
E _{2} \in {\mathcal S} _{2} ,
 +
$$
  
defines a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240439.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240440.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240441.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240442.png" />-finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240443.png" /> extends uniquely to a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240444.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240445.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240446.png" /> and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240447.png" /> are called, respectively, the product of the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240448.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240449.png" />, and the product of the measure spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240450.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240451.png" />. The completion of the product of the Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240452.png" /> and the Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240453.png" /> is the Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240454.png" />. Analogously one defines the product of an arbitrary finite number of measure spaces.
+
defines a measure on $  {\mathcal P} $;  
 +
if $  \mu _{1} $
 +
and $  \mu _{2} $
 +
are $  \sigma $-
 +
finite, $  \mu $
 +
extends uniquely to a measure on $  {\mathcal S} _{1} \times {\mathcal S} _{2} $,  
 +
denoted by $  \mu _{1} \times \mu _{2} $.  
 +
The measure $  \mu _{1} \times \mu _{2} $
 +
and the space $  (X _{1} \times X _{2} ,\  {\mathcal S} _{1} \times {\mathcal S} _{2} ,\  \mu _{1} \times \mu _{2} ) $
 +
are called, respectively, the product of the measures $  \mu _{1} $
 +
and $  \mu _{2} $,  
 +
and the product of the measure spaces $  (X _{1} ,\  {\mathcal S} _{1} ,\  \mu _{1} ) $
 +
and $  (X _{2} ,\  {\mathcal S} _{2} ,\  \mu _{2} ) $.  
 +
The completion of the product of the Lebesgue measure in $  \mathbf R^{k} $
 +
and the Lebesgue measure in $  \mathbf R^{l} $
 +
is the Lebesgue measure in $  \mathbf R^{k+l} $.  
 +
Analogously one defines the product of an arbitrary finite number of measure spaces.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240455.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240456.png" />, be an arbitrary family of measure spaces such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240457.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240458.png" />. The product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240459.png" /> is, by definition, the set of all functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240460.png" /> such that the value at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240461.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240462.png" />. A measurable rectangle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240463.png" /> is any set of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240464.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240465.png" /> and only finitely many sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240466.png" /> are different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240467.png" />. The family of measurable rectangles forms a semi-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240468.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240469.png" />-ring generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240470.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240471.png" /> and is called the product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240473.png" />-rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240474.png" />. Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240475.png" /> be the function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240476.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240477.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240478.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240479.png" /> thus defined is a measure which admits a unique extension to a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240480.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240481.png" />. The measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240482.png" /> is called the product of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240483.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240484.png" />.
+
Let $  (X _{i} ,\  {\mathcal S} _{i} ,\  \mu _{i} ) $,  
 +
$  i \in I $,  
 +
be an arbitrary family of measure spaces such that $  \mu _{i} (X _{i} ) = 1 $,  
 +
$  i \in I $.  
 +
The product space $  X = \prod _{ {i \in I}} X _{i} $
 +
is, by definition, the set of all functions on $  I $
 +
such that the value at each $  i \in I $
 +
is an element $  x _{i} \in X _{i} $.  
 +
A measurable rectangle in $  X $
 +
is any set of the form $  \prod _{ {i \in I}} E _{i} $,  
 +
where $  E _{i} \in {\mathcal S} _{i} $
 +
and only finitely many sets $  E _{i} $
 +
are different from $  X _{i} $.  
 +
The family of measurable rectangles forms a semi-ring $  {\mathcal P} $.  
 +
The $  \sigma $-
 +
ring generated by $  {\mathcal P} $
 +
is denoted by $  \prod _{ {i \in I}} {\mathcal S} _{i} $
 +
and is called the product of the $  \sigma $-
 +
rings $  {\mathcal S} _{i} $.  
 +
Now, let $  \mu $
 +
be the function on $  {\mathcal P} $
 +
defined by $  \mu (E) = \prod _{ {i \in I}} \mu _{i} (E _{i} ) $
 +
for $  E = \prod _{ {i \in I}} E _{i} $.  
 +
The function $  \mu $
 +
thus defined is a measure which admits a unique extension to a measure on $  \prod _{ {i \in I}} {\mathcal S} _{i} $,  
 +
denoted by $  \prod _{ {i \in I}} \mu _{i} $.  
 +
The measure space $  ( \prod _{ {i \in I}} X _{i} ,\  \prod _{ {i \in I}} {\mathcal S} _{i} ,\  \prod _{ {i \in I}} \mu _{i} ) $
 +
is called the product of the spaces $  (X _{i} ,\  {\mathcal S} _{i} ,\  \mu _{i} ) $,  
 +
$  i \in I $.
  
The product of an arbitrary number of measure spaces is a particular case of the following, more general, scheme, which plays an important role in probability theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240485.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240486.png" />, be a family of measurable spaces (each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240487.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240488.png" />-algebra), and suppose that for each finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240489.png" /> there is given a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240490.png" /> on the measurable spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240491.png" /> (the product of measures corresponds to the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240492.png" /> for all finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240493.png" />). Suppose further that each two measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240494.png" /> are compatible in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240495.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240496.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240497.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240498.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240499.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240500.png" /> (by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240501.png" /> is the mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240502.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240503.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240504.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240505.png" />). The following question arises: Is there a probability measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240506.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240507.png" /> for every finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240508.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240509.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240510.png" /> denotes the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240511.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240512.png" />? It turns out that such a measure does not always exist, and additional conditions must be imposed to guarantee its existence. One such condition is perfectness of the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240513.png" /> (corresponding to the one-point sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240514.png" />). The notion of a [[Perfect measure|perfect measure]] was first introduced by B.V. Gnedenko and A.N. Kolmogorov [[#References|[6]]]. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240515.png" /> with a totally-finite measure, as well as the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240516.png" /> itself, is called perfect if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240517.png" />-measurable real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240518.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240519.png" /> there is a Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240520.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240521.png" />. The perfectness assumption eliminates a series of  "pathological"  phenomena that arise in general measure theory.
+
The product of an arbitrary number of measure spaces is a particular case of the following, more general, scheme, which plays an important role in probability theory. Let $  (X _{i} ,\  {\mathcal S} _{i} ) $,  
 +
$  i \in I $,  
 +
be a family of measurable spaces (each $  {\mathcal S} _{i} $
 +
is a $  \sigma $-
 +
algebra), and suppose that for each finite subset $  I _{1} \subset  I $
 +
there is given a probability measure $  \mu _{ {I _ 1}} $
 +
on the measurable spaces $  ( \prod _{ {i \in I _ 1}} X _{i} ,\  \prod _{ {i \in I _ 1}} \in {\mathcal S} _{i} ) $(
 +
the product of measures corresponds to the case that $  \mu _{ {I _ 1}} = \prod _{ {i \in I _ 1}} \mu _{i} $
 +
for all finite $  I _{1} \subset  I $).  
 +
Suppose further that each two measures $  \mu _{ {I _ 1}} ,\  \mu _{ {I _ 2}} $
 +
are compatible in the sense that if $  I _{1} \subset  I _{2} $
 +
and $  p _{21} $
 +
is the projection of $  \prod _{ {i \in I _ 2}} X _{i} $
 +
onto $  \prod _{ {i \in I _ 1}} X _{i} $,  
 +
then $  \mu _{ {I _ 1}} (E) = \mu _{ {I _ 2}} p _ 21^{-1} (E) $
 +
for all $  E \in \prod _{ {i \in I _ 1}} {\mathcal S} _{i} $(
 +
by definition, $  p _{21} $
 +
is the mapping of $  \prod _{ {i \in I _ 2}} X _{i} $
 +
onto $  \prod _{ {i \in I _ 1}} X _{i} $
 +
such that $  (p _{21} (x )) _{i} = x _{i} $
 +
for all $  i \in I _{1} $).  
 +
The following question arises: Is there a probability measure on $  \prod _{ {i \in I}} {\mathcal S} _{i} $
 +
such that $  \mu _{ {I _ 1}} (E) = \mu p^{-1} (E) $
 +
for every finite $  I _{1} \subset  I $
 +
and every $  E \in \prod _{ {i \in I _ 1}} {\mathcal S} _{i} $,  
 +
where $  p $
 +
denotes the projection of $  \prod _{ {i \in I}} X _{i} $
 +
onto $  \prod _{ {i \in I _ 1}} X _{i} $?  
 +
It turns out that such a measure does not always exist, and additional conditions must be imposed to guarantee its existence. One such condition is perfectness of the measures $  \mu _{i} $(
 +
corresponding to the one-point sets $  i \in I $).  
 +
The notion of a [[Perfect measure|perfect measure]] was first introduced by B.V. Gnedenko and A.N. Kolmogorov [[#References|[6]]]. A space $  (X,\  {\mathcal S} ,\  \mu ) $
 +
with a totally-finite measure, as well as the measure $  \mu $
 +
itself, is called perfect if for every $  {\mathcal S} $-
 +
measurable real-valued function $  f $
 +
on $  X $
 +
there is a Borel set $  B \subset  f(X) $
 +
such that $  \mu (f ^ {\  -1} (B)) = \mu (X) $.  
 +
The perfectness assumption eliminates a series of  "pathological"  phenomena that arise in general measure theory.
  
 
==Measures in topological spaces.==
 
==Measures in topological spaces.==
The study of measures in topological spaces is usually concerned with measures defined on sets connected in some way or another with the topology of the underlying space. One of the typical approaches is the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240522.png" /> be an arbitrary topological space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240523.png" /> be the class of subsets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240524.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240525.png" /> is a continuous real-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240526.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240527.png" /> is a closed set. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240528.png" /> be the algebra generated by the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240529.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240530.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240531.png" />-algebra generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240532.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240533.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240535.png" />-algebra of Baire sets, cf. also [[Algebra of sets|Algebra of sets]]). Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240536.png" /> be the class of totally-finite finitely-additive measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240537.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240538.png" /> that are regular in the sense that
+
The study of measures in topological spaces is usually concerned with measures defined on sets connected in some way or another with the topology of the underlying space. One of the typical approaches is the following. Let $  X $
 +
be an arbitrary topological space and let $  {\mathcal Z} $
 +
be the class of subsets of the form $  f ^ {\  -1} (F \  ) $,  
 +
where $  f $
 +
is a continuous real-valued function on $  X $
 +
and $  F \subset  \mathbf R^{1} $
 +
is a closed set. Let $  \mathfrak A $
 +
be the algebra generated by the class $  {\mathcal Z} $
 +
and let $  {\mathcal B} $
 +
be the $  \sigma $-
 +
algebra generated by $  {\mathcal Z} $(
 +
$  {\mathcal B} $
 +
is called the $  \sigma $-
 +
algebra of Baire sets, cf. also [[Algebra of sets|Algebra of sets]]). Now let $  {\mathcal M} $
 +
be the class of totally-finite finitely-additive measures m $
 +
on $  \mathfrak A $
 +
that are regular in the sense that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240539.png" /></td> </tr></table>
+
$$
 +
m(E) \  = \  \sup \{ {m(Z)} : {Z \subset  E,\  Z \in {\mathcal Z}} \}
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240540.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240541.png" /> one distinguishes the subclasses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240542.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240543.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240544.png" /> formed by the (finitely-additive) measures possessing additional smoothness properties. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240545.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240546.png" /> for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240547.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240548.png" /> (this property is equivalent to the countable additivity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240549.png" />; the measures from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240550.png" /> admit unique extensions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240551.png" /> and hereafter it is assumed that they are given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240552.png" />); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240553.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240554.png" /> for every net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240555.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240556.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240557.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240558.png" /> there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240559.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240560.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240561.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240562.png" />.
+
for all $  E \in \mathfrak A $.  
 +
In $  {\mathcal M} $
 +
one distinguishes the subclasses $  {\mathcal M} _ \sigma  $,  
 +
$  {\mathcal M} _ \tau  $
 +
and $  {\mathcal M} _{t} $
 +
formed by the (finitely-additive) measures possessing additional smoothness properties. By definition, $  \mu \in {\mathcal M} _ \sigma  $
 +
if $  \mu (Z _{n} ) \downarrow 0 $
 +
for every sequence $  Z _{n} \downarrow \emptyset $,  
 +
$  Z _{n} \in {\mathcal Z} $(
 +
this property is equivalent to the countable additivity of $  \mu $;  
 +
the measures from $  {\mathcal M} _ \sigma  $
 +
admit unique extensions to $  {\mathcal B} $
 +
and hereafter it is assumed that they are given on $  {\mathcal B} $);  
 +
$  \mu \in {\mathcal M} _ \tau  $
 +
if $  \mu ( {\mathcal Z} _ \alpha  ) \downarrow 0 $
 +
for every net $  Z _ \alpha  \downarrow \emptyset $,  
 +
$  Z _ \alpha  \in {\mathcal Z} $;  
 +
and $  \mu \in {\mathcal M} _{t} $
 +
if for every $  \epsilon > 0 $
 +
there is a compact set $  K $
 +
such that $  \mu (E) < \epsilon $
 +
whenever $  E \subset  X\setminus K $,  
 +
$  E \in \mathfrak A $.
  
The inclusions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240563.png" /> hold. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240564.png" /> are called Baire measures.
+
The inclusions $  {\mathcal M} \supset {\mathcal M} _ \sigma  \supset {\mathcal M} _ \tau  \supset {\mathcal M} _{t} $
 +
hold. The elements of $  {\mathcal M} _ \sigma  $
 +
are called Baire measures.
  
There is an intimate connection between the measures belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240565.png" /> and the linear functionals on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240566.png" /> of bounded continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240567.png" />. Namely, the formula
+
There is an intimate connection between the measures belonging to $  {\mathcal M} $
 +
and the linear functionals on the space $  C(X) $
 +
of bounded continuous functions on $  X $.  
 +
Namely, the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240568.png" /></td> </tr></table>
+
$$
 +
\Lambda (f \  ) \  = \  \int\limits _ { X } f \  dm
 +
$$
  
establishes a one-to-one correspondence between the finitely-additive measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240569.png" /> and the non-negative linear functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240570.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240571.png" /> (non-negative means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240572.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240573.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240574.png" />). Moreover, for every set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240575.png" />,
+
establishes a one-to-one correspondence between the finitely-additive measures m \in {\mathcal M} $
 +
and the non-negative linear functionals $  \Lambda $
 +
on $  C(X) $(
 +
non-negative means that $  \Lambda (f \  ) \geq 0 $
 +
whenever $  f(x) \geq 0 $,  
 +
$  x \in X $).  
 +
Moreover, for every set $  Z \in {\mathcal Z} $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240576.png" /></td> </tr></table>
+
$$
 +
m(Z) \  = \  \mathop{\rm inf}\nolimits \{ {\Lambda (f \  )} : {\chi _{Z} \leq  f \leq  1
 +
} \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240577.png" /> is the indicator function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240578.png" />. This correspondence takes the measures from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240579.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240581.png" />-smooth functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240582.png" /> (i.e. functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240583.png" /> with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240584.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240585.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240586.png" />), the measures from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240587.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240589.png" />-smooth functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240590.png" /> (i.e. functionals such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240591.png" /> for every net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240592.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240593.png" />), and the measures from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240594.png" /> into dense functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240595.png" /> (i.e. with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240596.png" /> for every net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240597.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240598.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240599.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240600.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240601.png" /> uniformly on compact subsets; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240602.png" /> is the uniform norm).
+
where $  \chi _{Z} $
 +
is the indicator function of $  Z $.  
 +
This correspondence takes the measures from $  {\mathcal M} _ \sigma  $
 +
into $  \sigma $-
 +
smooth functionals $  \Lambda $(
 +
i.e. functionals $  \Lambda $
 +
with the property that $  \Lambda (f _{n} ) \rightarrow 0 $
 +
if $  f _{n} \downarrow 0 $
 +
in $  C(X) $),  
 +
the measures from $  {\mathcal M} _ \tau  $
 +
into $  \tau $-
 +
smooth functionals $  \Lambda $(
 +
i.e. functionals such that $  \Lambda (f _ \alpha  ) \rightarrow 0 $
 +
for every net $  f _ \alpha  \downarrow 0 $
 +
in $  C(X) $),  
 +
and the measures from $  {\mathcal M} _{t} $
 +
into dense functionals $  \Lambda $(
 +
i.e. with the property that $  \Lambda (f _ \alpha  ) \rightarrow 0 $
 +
for every net $  f _ \alpha  $
 +
in $  C(X) $
 +
such that $  \| f _ \alpha  \| \leq  1 $
 +
for all $  \alpha $
 +
and $  f _ \alpha  \rightarrow 0 $
 +
uniformly on compact subsets; here $  \| \cdot \| $
 +
is the uniform norm).
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240603.png" /> is usually endowed with the weak topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240604.png" />, in which a basis of neighbourhoods consists of the sets of the form
+
The space $  {\mathcal M} $
 +
is usually endowed with the weak topology $  w $,  
 +
in which a basis of neighbourhoods consists of the sets of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240605.png" /></td> </tr></table>
+
$$
 +
U(m _{0} ; \  f _{1} \dots f _{n} ,\  \epsilon )\  =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240606.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left \{ m : \  \left | \int\limits _ { X } f _{k} \  (dm-dm _{0} ) \right | < \epsilon ,\  k = 1 \dots
 +
n,\  f _{1} \dots f _{n} \in C(X) \right \} .
 +
$$
  
With the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240608.png" /> is a completely-regular Hausdorff space. Convergence in the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240609.png" /> is usually denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240610.png" />. For the convergence of a net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240611.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240612.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240613.png" />, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240614.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240615.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240616.png" />. Another necessary and sufficient condition for the convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240617.png" /> is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240618.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240619.png" /> such that there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240620.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240621.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240622.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240623.png" />. If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240624.png" /> is completely regular and Hausdorff, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240625.png" /> is metrizable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240626.png" /> is metrizable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240627.png" /> is metrizable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240628.png" /> admits a metric in which it is separable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240629.png" /> is separable, and it admits a metric in which it is complete if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240630.png" /> has a complete metric. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240631.png" /> is metrizable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240632.png" /> is metrizable if and only if it is metrizable by the [[Lévy–Prokhorov metric|Lévy–Prokhorov metric]].
+
With the topology $  w $,  
 +
$  {\mathcal M} $
 +
is a completely-regular Hausdorff space. Convergence in the topology $  w $
 +
is usually denoted by the symbol $  \Rightarrow $.  
 +
For the convergence of a net m _ \alpha  $
 +
to $  m $:  
 +
m _ \alpha  \Rightarrow m $,  
 +
it is necessary and sufficient that $  m _ \alpha  (X) \rightarrow m(X) $
 +
and $  \lim\limits \  \sup \  m _ \alpha  (Z) \leq  m(Z) $
 +
for all $  Z \in {\mathcal Z} $.  
 +
Another necessary and sufficient condition for the convergence $  m _ \alpha  \Rightarrow m $
 +
is that $  m _ \alpha  (E) \rightarrow m(E) $
 +
for all $  E \in \mathfrak A $
 +
such that there are $  Z _{1} ,\  Z _{2} \in {\mathcal Z} $
 +
with $  X \setminus  E \subset  Z _{1} $,  
 +
$  E \subset  Z _{2} $,  
 +
and $  m(Z _{1} \cap Z _{2} ) = 0 $.  
 +
If the space $  X $
 +
is completely regular and Hausdorff, then $  {\mathcal M} _ \tau  $
 +
is metrizable if and only if $  X $
 +
is metrizable. If $  X $
 +
is metrizable, then $  {\mathcal M} _ \tau  $
 +
admits a metric in which it is separable if and only if $  X $
 +
is separable, and it admits a metric in which it is complete if and only if $  X $
 +
has a complete metric. If $  X $
 +
is metrizable, then $  {\mathcal M} _ \sigma  $
 +
is metrizable if and only if it is metrizable by the [[Lévy–Prokhorov metric|Lévy–Prokhorov metric]].
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240633.png" /> is sequentially closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240634.png" /> (Aleksandrov's theorem). A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240635.png" /> is called tight if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240636.png" /> and if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240637.png" /> there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240638.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240639.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240640.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240641.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240642.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240643.png" /> is tight, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240644.png" /> is relatively compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240645.png" />; conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240646.png" /> is metrizable and topologically complete, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240647.png" /> is relatively compact, and if every measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240648.png" /> is concentrated on some separable subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240649.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240650.png" /> is tight (Prokhorov's theorem).
+
The space $  {\mathcal M} _ \sigma  $
 +
is sequentially closed in $  {\mathcal M} $(
 +
Aleksandrov's theorem). A set $  A \subset  {\mathcal M} $
 +
is called tight if $  \sup \{ {m(X)} : {m \in A} \} < \infty $
 +
and if for every $  \epsilon > 0 $
 +
there is a compact set $  K $
 +
such that $  m(E) < \epsilon $
 +
for all $  E \subset  X\setminus K $,  
 +
m \in A $
 +
and $  E \in \mathfrak A $.  
 +
If $  A \subset  {\mathcal M} _ \sigma  $
 +
is tight, then $  A $
 +
is relatively compact in $  {\mathcal M} _ \sigma  $;  
 +
conversely, if $  X $
 +
is metrizable and topologically complete, then $  A \subset  {\mathcal M} _ \sigma  $
 +
is relatively compact, and if every measure in $  A $
 +
is concentrated on some separable subset of $  X $,  
 +
then $  A $
 +
is tight (Prokhorov's theorem).
  
Under certain conditions the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240651.png" /> can be extended to Borel measures, i.e. measures defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240652.png" />-algebra of Borel sets (see [[Borel set|Borel set]]; [[Borel measure|Borel measure]]). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240653.png" /> is a normal countably-paracompact Hausdorff space, then every measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240654.png" /> admits a unique extension to a regular Borel measure. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240655.png" /> is completely regular and Hausdorff, then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240656.png" />-smooth (tight) Baire measure admits a unique extension to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240657.png" />-smooth (tight) Borel measure.
+
Under certain conditions the elements of $  {\mathcal M} _ \sigma  $
 +
can be extended to Borel measures, i.e. measures defined on the $  \sigma $-
 +
algebra of Borel sets (see [[Borel set|Borel set]]; [[Borel measure|Borel measure]]). For example, if $  X $
 +
is a normal countably-paracompact Hausdorff space, then every measure $  \mu \in {\mathcal M} _ \sigma  $
 +
admits a unique extension to a regular Borel measure. If $  X $
 +
is completely regular and Hausdorff, then every $  \tau $-
 +
smooth (tight) Baire measure admits a unique extension to a $  \tau $-
 +
smooth (tight) Borel measure.
  
The support of a Baire (Borel) measure is the smallest set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240658.png" /> (respectively, the smallest closed set) the measure of which is equal to the measure of the whole space. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240659.png" />-smooth measure has a support.
+
The support of a Baire (Borel) measure is the smallest set $  Z \in {\mathcal Z} $(
 +
respectively, the smallest closed set) the measure of which is equal to the measure of the whole space. Every $  \tau $-
 +
smooth measure has a support.
  
Often, when measures in topological spaces (especially in locally compact Hausdorff spaces) are considered, it is assumed that the Borel and Baire measures are given on less-wide classes of sets, more precisely — on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240660.png" />-rings generated by compact sets and, respectively, compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240661.png" />-sets.
+
Often, when measures in topological spaces (especially in locally compact Hausdorff spaces) are considered, it is assumed that the Borel and Baire measures are given on less-wide classes of sets, more precisely — on $  \sigma $-
 +
rings generated by compact sets and, respectively, compact $  G _ \delta  $-
 +
sets.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240662.png" /> be a locally compact Hausdorff topological group. A left Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240663.png" /> is a measure defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240664.png" />-ring generated by all compact subsets that does not vanish identically and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240665.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240666.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240667.png" /> in the domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240668.png" />. A right Haar measure is defined in the same manner but with the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240669.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240670.png" />. On any group of the type considered a left Haar measure exists and is unique (up to a multiplicative positive constant). Every left Haar measure is regular in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240671.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240672.png" /> are compact sets. The right Haar measure has analogous properties. The Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240673.png" /> is a particular case of the [[Haar measure|Haar measure]]. See also [[Measure in a topological vector space|Measure in a topological vector space]].
+
Let $  G $
 +
be a locally compact Hausdorff topological group. A left Haar measure on $  G $
 +
is a measure defined on the $  \sigma $-
 +
ring generated by all compact subsets that does not vanish identically and is such that $  \mu (xE) = \mu (E) $
 +
for all $  x \in G $
 +
and $  E $
 +
in the domain of $  \mu $.  
 +
A right Haar measure is defined in the same manner but with the condition $  \mu (xE) = \mu (E) $
 +
replaced by $  \mu (Ex) = \mu (E) $.  
 +
On any group of the type considered a left Haar measure exists and is unique (up to a multiplicative positive constant). Every left Haar measure is regular in the sense that $  \mu (E) = \sup \{ {\mu (K)} : {K \subset  E} \} $,  
 +
where $  K $
 +
are compact sets. The right Haar measure has analogous properties. The Lebesgue measure on $  \mathbf R^{k} $
 +
is a particular case of the [[Haar measure|Haar measure]]. See also [[Measure in a topological vector space|Measure in a topological vector space]].
  
 
==Isomorphism of measure spaces.==
 
==Isomorphism of measure spaces.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240674.png" /> be a measure space. Call two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240675.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240677.png" />-equal (written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240678.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240679.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240680.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240681.png" /> denotes the symmetric difference of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240682.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240683.png" />, cf. [[Symmetric difference of sets|Symmetric difference of sets]]). Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240684.png" /> the class of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240685.png" /> with this equality relation. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240686.png" /> the set-theoretic operations, performed a finite (or countable) number of times are correctly defined: for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240687.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240688.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240689.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240690.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240691.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240692.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240693.png" /> is carried over, in an obvious manner, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240694.png" />.
+
Let $  (X,\  {\mathcal S} ,\  \mu ) $
 +
be a measure space. Call two sets $  E,\  E^ \prime  \in {\mathcal S} $
 +
$  \mu $-
 +
equal (written $  E = E^ \prime  $
 +
$  [ \mu ] $)  
 +
if $  \mu (E \Delta E^ \prime  ) = 0 $(
 +
where $  E \Delta E^ \prime  $
 +
denotes the symmetric difference of $  E $
 +
and $  E^ \prime  $,  
 +
cf. [[Symmetric difference of sets|Symmetric difference of sets]]). Denote by $  {\mathcal S} _ \mu  $
 +
the class of sets $  {\mathcal S} $
 +
with this equality relation. In $  {\mathcal S} _ \mu  $
 +
the set-theoretic operations, performed a finite (or countable) number of times are correctly defined: for example, if $  E _{1} = E _ 1^ \prime  $
 +
$  [ \mu ] $
 +
and $  E _{2} = E _ 2^ \prime  $
 +
$  [ \mu ] $,  
 +
then $  E _{1} \cup E _{2} = E _ 1^ \prime  \cup E _ 2^ \prime  $
 +
$  [ \mu ] $.  
 +
The measure $  \mu $
 +
is carried over, in an obvious manner, to $  {\mathcal S} _ \mu  $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240695.png" /> be the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240696.png" /> consisting of the sets of finite measure. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240697.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240698.png" /> is a metric. The measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240699.png" /> is said to be separable if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240700.png" /> with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240701.png" /> is separable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240702.png" /> is a space with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240703.png" />-finite measure and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240704.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240705.png" /> is countably generated (i.e. there is a countable family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240706.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240707.png" /> is the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240708.png" />-ring that contains this family), then the metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240709.png" /> is separable.
+
Let $  \widetilde{ {\mathcal S} }  _ \mu  $
 +
be the subset of $  {\mathcal S} _ \mu  $
 +
consisting of the sets of finite measure. The function $  \rho (E,\  E^ \prime  ) = \mu (E \Delta E^ \prime  ) $
 +
on $  \widetilde{ {\mathcal S} }  _ \mu  \times \widetilde{ {\mathcal S} }  _ \mu  $
 +
is a metric. The measure space $  (X,\  {\mathcal S} ,\  \mu ) $
 +
is said to be separable if the space $  \widetilde{ {\mathcal S} }  _ \mu  $
 +
with metric $  \rho $
 +
is separable. If $  (X,\  {\mathcal S} ,\  \mu ) $
 +
is a space with a $  \sigma $-
 +
finite measure and the $  \sigma $-
 +
ring $  {\mathcal S} $
 +
is countably generated (i.e. there is a countable family $  \{ E _{n} \} \subset  {\mathcal S} $
 +
such that $  {\mathcal S} $
 +
is the smallest $  \sigma $-
 +
ring that contains this family), then the metric space $  \widetilde{ {\mathcal S} }  _ \mu  $
 +
is separable.
  
Two measure spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240710.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240711.png" /> are said to be isomorphic if there is a one-to-one mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240712.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240713.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240714.png" /> such that
+
Two measure spaces, $  (X _{1} ,\  {\mathcal S} _{1} ,\  \mu _{1} ) $
 +
and $  (X _{2} ,\  {\mathcal S} _{2} ,\  \mu _{2} ) $
 +
are said to be isomorphic if there is a one-to-one mapping $  \phi $
 +
of $  ( {\mathcal S} _{1} ) _{ {\mu _ 1}} $
 +
onto $  ( {\mathcal S} _{2} ) _{ {\mu _ 2}} $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240715.png" /></td> </tr></table>
+
$$
 +
\phi (E\setminus F \  ) \  = \
 +
\phi (E) \setminus  \phi (F \  ) ,\ \
 +
\phi (E \cup F \  ) \  = \  \phi (E) \cup \phi (F \  )
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240716.png" /></td> </tr></table>
+
$$
 +
\mu _{1} (E) \  = \  \mu _{2} ( \phi (E)) \ \
 +
\textrm{ for \  all } \  E,\  F \in ( {\mathcal S} _{1} )
 +
_{ {\mu _ 1}} .
 +
$$
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240717.png" /> be an arbitrary space with a totally-finite measure. There is a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240718.png" /> into disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240719.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240720.png" /> such that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240721.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240722.png" /> is isomorphic either to a measure concentrated at one point or to a measure which is equal, up to a positive factor, to the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240723.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240724.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240725.png" />, and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240726.png" /> may have arbitrary cardinality (the Maharan–Kolmogorov theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240727.png" /> is separable, non-atomic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240728.png" />, then it is isomorphic to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240729.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240730.png" /> countable, which in turn is isomorphic to the unit interval with the Lebesgue measure.
+
Now, let $  (X,\  {\mathcal S} ,\  \mu ) $
 +
be an arbitrary space with a totally-finite measure. There is a partition of $  X $
 +
into disjoint sets $  X _{n} \in {\mathcal S} $,
 +
$  n = 1,\  2 \dots $
 +
such that the restriction of $  \mu $
 +
to $  X _{n} $
 +
is isomorphic either to a measure concentrated at one point or to a measure which is equal, up to a positive factor, to the direct product $  \prod _{ {i \in I}} (U _{i} ,\  {\mathcal U} _{i} ,\  u _{i} ) $,  
 +
where $  U _{i} = \{ 0,\  1 \} $,
 +
$  u _{i} ( \{ 0 \} ) = u _{i} ( \{ 1 \} ) = 1/2 $,  
 +
and the set $  I $
 +
may have arbitrary cardinality (the Maharan–Kolmogorov theorem). If $  (X,\  {\mathcal S} ,\  \mu ) $
 +
is separable, non-atomic and $  \mu (X) = 1 $,  
 +
then it is isomorphic to the space $  \prod _{ {i \in I}} (U _{i} ,\  {\mathcal U} _{i} ,\  u _{i} ) $
 +
with $  I $
 +
countable, which in turn is isomorphic to the unit interval with the Lebesgue measure.
  
 
Side by side with the theory of measures regarded as functions on subsets of some set, the theory of measures as functions on the elements of a Boolean ring (or on a [[Boolean algebra|Boolean algebra]]) has been developed; these theories are in many respects parallel. Another widespread construction of measures goes back to W. Young and P. Daniell (see [[#References|[12]]]). Theories dealing with measures with real or complex values, or with values belonging to some algebraic structure, were developed in addition to the theory of positive measures.
 
Side by side with the theory of measures regarded as functions on subsets of some set, the theory of measures as functions on the elements of a Boolean ring (or on a [[Boolean algebra|Boolean algebra]]) has been developed; these theories are in many respects parallel. Another widespread construction of measures goes back to W. Young and P. Daniell (see [[#References|[12]]]). Theories dealing with measures with real or complex values, or with values belonging to some algebraic structure, were developed in addition to the theory of positive measures.
Line 278: Line 824:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)  {{MR|0167578}} {{ZBL|1196.28001}} {{ZBL|0017.30004}}  {{ZBL|63.0183.05}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)  {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)  {{MR|0117523}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)  {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Neveu,  "Mathematical foundations of the calculus of probabilities" , Holden-Day  (1965)  (Translated from French)  {{MR|0198505}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.V. Gnedenko,  A.N. Kolmogorov,  "Limit distributions for sums of independent random variables" , Addison-Wesley  (1954)  (Translated from Russian)  {{MR|0062975}} {{ZBL|0056.36001}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.S. Varadarajan,  "Measures on topological spaces"  ''Mat. Sb.'' , '''55''' :  1  (1961)  pp. 35–100  (In Russian)  {{MR|0148838}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  K.R. Parthasarathy,  "Probability measures on metric spaces" , Acad. Press  (1967)  {{MR|0226684}} {{ZBL|0153.19101}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P. Billingsley,  "Convergence of probability measures" , Wiley  (1968)  {{MR|0233396}} {{ZBL|0172.21201}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  R. Sikorski,  "Boolean algebras" , Springer  (1969)  {{MR|0249336}} {{MR|0242724}} {{ZBL|0191.31505}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  D.A. Vladimirov,  "Boolesche Algebren" , Akademie Verlag  (1978)  (Translated from Russian)  {{MR|0524392}} {{ZBL|0385.06018}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)  {{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  J. Diestel,  J.J. Uhl jr.,  "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc.  (1977)  {{MR|0453964}} {{ZBL|0369.46039}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)  {{MR|0167578}} {{ZBL|1196.28001}} {{ZBL|0017.30004}}  {{ZBL|63.0183.05}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)  {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)  {{MR|0117523}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)  {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Neveu,  "Mathematical foundations of the calculus of probabilities" , Holden-Day  (1965)  (Translated from French)  {{MR|0198505}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.V. Gnedenko,  A.N. Kolmogorov,  "Limit distributions for sums of independent random variables" , Addison-Wesley  (1954)  (Translated from Russian)  {{MR|0062975}} {{ZBL|0056.36001}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.S. Varadarajan,  "Measures on topological spaces"  ''Mat. Sb.'' , '''55''' :  1  (1961)  pp. 35–100  (In Russian)  {{MR|0148838}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  K.R. Parthasarathy,  "Probability measures on metric spaces" , Acad. Press  (1967)  {{MR|0226684}} {{ZBL|0153.19101}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P. Billingsley,  "Convergence of probability measures" , Wiley  (1968)  {{MR|0233396}} {{ZBL|0172.21201}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  R. Sikorski,  "Boolean algebras" , Springer  (1969)  {{MR|0249336}} {{MR|0242724}} {{ZBL|0191.31505}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  D.A. Vladimirov,  "Boolesche Algebren" , Akademie Verlag  (1978)  (Translated from Russian)  {{MR|0524392}} {{ZBL|0385.06018}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)  {{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  J. Diestel,  J.J. Uhl jr.,  "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc.  (1977)  {{MR|0453964}} {{ZBL|0369.46039}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Line 286: Line 830:
 
The procedure for extending a measure, as described under the heading  "Extension of measures" , is due to C. Carathéodory, and one often speaks of Carathéodory extension, with the accompanying phrases Carathéodory extension theorem and Carathéodory outer (inner) measure (cf. [[Carathéodory measure|Carathéodory measure]]).
 
The procedure for extending a measure, as described under the heading  "Extension of measures" , is due to C. Carathéodory, and one often speaks of Carathéodory extension, with the accompanying phrases Carathéodory extension theorem and Carathéodory outer (inner) measure (cf. [[Carathéodory measure|Carathéodory measure]]).
  
Recall that a ring (respectively, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240731.png" />-ring) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240732.png" /> of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240733.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240734.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240735.png" />, is called a Boolean algebra or an algebra (respectively, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240737.png" />-algebra or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240739.png" />-field, cf. also [[Algebra of sets|Algebra of sets]]). Usually, in a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240740.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240741.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240742.png" /> can be proved to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240743.png" />-field (this holds, in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240744.png" />).
+
Recall that a ring (respectively, a $  \sigma $-
 +
ring) $  {\mathcal A} $
 +
of subsets of a set $  X $
 +
such that $  A \in {\mathcal A} $
 +
implies $  X \setminus  A \in {\mathcal A} $,  
 +
is called a Boolean algebra or an algebra (respectively, a $  \sigma $-
 +
algebra or a $  \sigma $-
 +
field, cf. also [[Algebra of sets|Algebra of sets]]). Usually, in a measure space $  ( X ,\  {\mathcal S} ,\  \mu ) $
 +
the $  \sigma $-
 +
ring $  {\mathcal S} $
 +
can be proved to be a $  \sigma $-
 +
field (this holds, in particular, if $  \mu ( X ) < \infty $).
  
 
The phrase  "totally (s-) finite"  is seldom used.
 
The phrase  "totally (s-) finite"  is seldom used.
  
Borel has given very nice ideas in order to construct the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240745.png" />, but Lebesgue was the first to give a satisfactory construction of it, as a byproduct of the construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240746.png" />.
+
Borel has given very nice ideas in order to construct the measure $  \lambda^ \prime  $,  
 +
but Lebesgue was the first to give a satisfactory construction of it, as a byproduct of the construction of $  \overline \lambda \; $.
  
A product space is also often written as a (kind of) tensor product: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240747.png" />.
+
A product space is also often written as a (kind of) tensor product: $  ( X _{1} \times X _{2} ,\  {\mathcal S} _{1} \otimes {\mathcal S} _{2} ,\  \mu _{1} \otimes \mu _{2} ) $.
  
A family of measurable spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240748.png" /> with compatible probability measures on each finite product is called a projective system of measure spaces, and the corresponding probability measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240749.png" />, if it exists, is called the projective limit; it exists if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240750.png" /> is countable (the Ionescu–Tulcea theorem, cf. [[#References|[5]]]).
+
A family of measurable spaces $  ( X _{i} ,\  {\mathcal S} _{i} ) _{i} $
 +
with compatible probability measures on each finite product is called a projective system of measure spaces, and the corresponding probability measure on $  \prod {X _ i} $,  
 +
if it exists, is called the projective limit; it exists if $  I $
 +
is countable (the Ionescu–Tulcea theorem, cf. [[#References|[5]]]).
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240751.png" /> is a topological space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240752.png" /> is its Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240753.png" />-field; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240754.png" /> is perfect for every finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240755.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240756.png" /> is a Polish space or, more generally, a Luzin space (in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240757.png" /> is often called a standard measurable space) or, still more generally, a Suslin space (in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240758.png" /> is sometimes called a Blackwell measurable space) (cf. (the editorial comments to) [[Descriptive set theory|Descriptive set theory]]).
+
Suppose that $  X $
 +
is a topological space and $  {\mathcal S} $
 +
is its Borel $  \sigma $-
 +
field; then $  ( X ,\  {\mathcal S} ,\  \mu ) $
 +
is perfect for every finite measure $  \mu $
 +
if $  X $
 +
is a Polish space or, more generally, a Luzin space (in which case $  ( X ,\  {\mathcal S} ) $
 +
is often called a standard measurable space) or, still more generally, a Suslin space (in which case $  ( X ,\  {\mathcal S} ) $
 +
is sometimes called a Blackwell measurable space) (cf. (the editorial comments to) [[Descriptive set theory|Descriptive set theory]]).
  
The converse part of Prokhorov's theorem is not true when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240759.png" /> is the space of rational numbers, or, more generally, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240760.png" /> is a [[Luzin space|Luzin space]] which is not Polish. See [[#References|[a1]]].
+
The converse part of Prokhorov's theorem is not true when $  X $
 +
is the space of rational numbers, or, more generally, when $  X $
 +
is a [[Luzin space|Luzin space]] which is not Polish. See [[#References|[a1]]].
  
In the abstract setting, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240761.png" /> is a sequence of finite measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240762.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240763.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240764.png" />-field, such that
+
In the abstract setting, whenever $  ( \mu _{n} ) $
 +
is a sequence of finite measures on $  ( X ,\  {\mathcal S} ) $,
 +
where $  {\mathcal S} $
 +
is a $  \sigma $-
 +
field, such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240765.png" /></td> </tr></table>
+
$$
 +
m (A) \  = \  \lim\limits _ { n } \  \mu _{n} (A)
 +
$$
  
exists for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240766.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063240/m063240767.png" /> is also a measure (the Vitali–Hahn–Saks theorem, cf. [[#References|[3]]] or [[#References|[5]]]).
+
exists for any $  A \in {\mathcal S} $,  
 +
then m $
 +
is also a measure (the Vitali–Hahn–Saks theorem, cf. [[#References|[3]]] or [[#References|[5]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Preiss,  "Metric spaces in which Prokhorov's theorem is not valid"  ''Z. Wahrscheinlichkeitstheor. Verw. Gebiete'' , '''27'''  (1973)  pp. 109–116  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Cohn,  "Measure theory" , Birkhäuser  (1980)  {{MR|0578344}} {{ZBL|0436.28001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''I''' , Springer  (1979)  {{MR|0551496}} {{ZBL|0416.43001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)  {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Preiss,  "Metric spaces in which Prokhorov's theorem is not valid"  ''Z. Wahrscheinlichkeitstheor. Verw. Gebiete'' , '''27'''  (1973)  pp. 109–116  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Cohn,  "Measure theory" , Birkhäuser  (1980)  {{MR|0578344}} {{ZBL|0436.28001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''I''' , Springer  (1979)  {{MR|0551496}} {{ZBL|0416.43001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)  {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR></table>

Latest revision as of 23:21, 1 February 2020


measure of a set

A notion that generalizes those of the length of segments, the area of figures and the volume of bodies, and that corresponds intuitively to the mass of a set for some mass distribution throughout the space. The notion of the measure of a set arose in the theory of functions of a real variable in connection with the study and improvement of the notion of an integral.

Definition and general properties.

Let $X$ be a set and let $\mathcal{E}$ be a class of subsets of $X$. A non-negative (not necessarily finite) set function $\lambda$ defined on $\mathcal{E}$ is called additive, finitely additive or countably additive if

\[\lambda \left( \bigcup\limits_{i = 1}^n E_i \right) = \sum\limits_{i = 1}^n {\lambda ({E_i})} \]

whenever

\[ E_i \in \mathcal{E}, \quad \bigcup\limits_{i=1}^n E_i\in \mathcal{E}, \quad E_i \bigcap E_j=\emptyset, i \ne j, \]

for, respectively, $n=2$, $n$ arbitrary finite, and $n \le \infty$.

A collection $\mathcal{P}$ of subsets of $X$ is called a semi-ring of sets if

  1. $ \emptyset \in \mathcal{P} $
  2. $ E_1, E_2 \in \mathcal{P} \implies E_1 \cap E_2 \in \mathcal{P} $
  3. $ E, E_1 \in \mathcal{P}, E_1 \subset E \implies E$ is representable as $ E = \bigcup_{i=1}^n E_i, E_i \cap E_j = \emptyset $ for $i \ne j, E_i \in \mathcal{P}, i = 1 \dots n, n < \infty$ .

A collection $ \mathcal{R}$ of subsets of $X$ is called a ring of sets if

  1. $\emptyset \in \mathcal{R}$
  2. $E_1, E_2 \in \mathcal{R} \implies E_1 \cup E_2 \in \mathcal{R}, E_1 \setminus E_2 \in \mathcal{R}$.

An example of a semi-ring is: $X = \mathbf{R}^k $, $\mathcal{P}$ is the collection of all intervals of the form

\[ \{x = (x_1, \dots, x_k) \in \mathbf{R}^k \mid a_i \le x_i < b_i, i = 1, \dots, k\} \]

where $ a_i, b_i \in \mathbf{R} $ for $ i = 1, \dots, k $. The collection of all possible finite unions of such intervals is a ring.

A collection $\mathcal{S}$ of subsets of $X$ is called a $\sigma$-ring if

  1. $\emptyset \in \mathcal{S} $
  2. $E_1, E_2 \in \mathcal{S} \implies E_1 \setminus E_2 \in \mathcal{S}$
  3. $E_i \in \mathcal{S} \quad (i = 1, 2, \dots) \implies \bigcup_{i=1}^\infty E_i \in S$.

Every $\sigma$-ring is a ring; every ring is a semi-ring.

A finitely-additive measure is a non-negative finitely-additive set function $m$ such that $m(\emptyset) = 0$. The domain of definition $\mathcal{E}_m$ of a finitely-additive measure may be a semi-ring, a ring or a $\sigma$-ring. In the definition of a finitely-additive measure on a ring or on a $\sigma$-ring the condition of finite additivity can be weakened to additivity, which leads to the same notion.

If $m$ is a finitely-additive measure, if the sets $E, E_1, \dots, E_n$ belong to its domain of definition, and if $E \subset \bigcup_{i=1}^n E_i$ , then

\[ m(E) \le \sum_{i=1}^n m(E_i). \]

Let $m_1$ be a finitely-additive measure with domain $\mathcal{E}_{m_1}$. A finitely-additive measure $m_2$ with domain $\mathcal{E}_{m_2}$ is called an extension of $m_1$ if $\mathcal{E}_{m_1} \subset \mathcal{E}_{m_2}$ and $m_2(E) = m_1(E) \quad \forall E\in \mathcal{E}_{m_1}$.

Every finitely-additive measure $m$ defined on a semi-ring $\mathcal{P}$ admits a unique extension to a finitely-additive measure $m'$ on the smallest ring $\mathcal{R}(\mathcal{P})$ containing $\mathcal{P}$. This extension is defined as follows: Every $E \in \mathcal{R}(\mathcal{P})$ is representable as $E = \bigcup_{i=1}^n E_i, E_i \in \mathcal{P}, E_i \cap E_j = \emptyset, i \ne j$, and one sets

\[ m'(E) = \sum_{i=1}^n m(E_i). \]

A finitely-additive measure that has the property of countable additivity is called a measure. Examples of measures: Let $X$ be an arbitrary non-empty set, let $\mathcal{E}_\mu$ be a $ \sigma $-ring, a ring or a semi-ring of subsets of $ X $, let $ \{x_1, x_2, \dots \} $ be a countable subset of $ X $, and let $ p_1, p_2, \dots $ be non-negative numbers. Then the function

\[ \mu(E) = \sum_n p_n \delta_{x_n}(E), \] where $ \delta_x (E) = 1 $ if $ x \in E $ and $ \delta_x(E) = 0 $ if $ x \notin E $, is a measure defined on $ \mathcal{E}_\mu $. The measures $ \delta_x $ are called elementary, degenerate or Dirac measures (sometimes, Dirac masses). Not every finitely-additive measure is a measure. For example, if $ X $ is the set of rational points of the segment $ [0,1] $, $ \mathcal{P} $ is the semi-ring of all possible intersections of subintervals of $ [0,1] $ with $ X $, and for every $ a, b $, $ 0\le a\le b\le 1 $,

\[ m((a, b) \cap X) = m([a, b) \cap X) = m((a, b] \cap X) = m([a,b] \cap X) = b - a, \]

then $ m $ is finitely additive, but not countably additive on $ \mathcal{P} $.

A (finitely-additive) measure $ m $ with domain $ \mathcal{E}_m $ is said to be finite (respectively, $ \sigma $-finite) if $ m(E) < \infty $ for all $ E \in \mathcal{E}_m $ (respectively, if for every $ E\in \mathcal{E}_m $ there is a sequence of sets $ \{E_i\} $ in $ \mathcal{E}_m $ such that $ E \subset \bigcup_{i=1}^\infty E_i $ and $ m(E_i)< \infty $, $ i = 1, 2, \dots $).

A (finitely-additive) measure $ m $ is said to be totally finite (totally $ \sigma $-finite) if it is finite (respectively, $ \sigma $-finite) and $ X \in \mathcal{E}_m $.

A pair $ (X, \mathcal{S}) $, where $ X $ is a set and $ \mathcal{S} $ is a $ \sigma $-ring of subsets of $ X $ such that $ \bigcup_{E \in \mathcal{S}} E = X$, is called a measurable space. A triple $ (X, \mathcal{S}, \mu) $, where $ (X, \mathcal{S}) $ is a measurable space and $ \mu $ is a measure on $ \mathcal{S} $, is called a measure space. A space with a totally-finite measure $ \mu $ normalized by the condition $ \mu(X) = 1 $ is called a probability space. In abstract measure theory, where the basic notions are a measurable space $ (X, \mathcal{S}) $ or a measure space $ (X, \mathcal{S}, \mu) $, the elements of $ \mathcal{S} $ are also referred to as measurable sets (cf. also Measurable set).

Properties of measure spaces.

Let $\{E_i\}$ be an arbitrary sequence of measurable sets. Then

  1. $ \mu(\lim\inf_{i\to\infty} E_i)\le \lim\inf_{i \to \infty} \mu(E_i) $
  2. if $ \mu(\bigcup_{i=i_0}^\infty E_i) < \infty $ for some $i_0$, then \[\mu\left( \limsup\limits_{i\to\infty} E_i\right) \ge \limsup\limits_{i\to \infty} \mu(E_i)\]
  3. if $\lim_{i \to \infty} E_i$ exists and the condition in 2) is satisfied, then

\[ \mu\left( \lim\limits_{i\to\infty} E_i\right) = \lim\limits_{i\to \infty} \mu(E_i) \]

A finitely-additive measure $m$ defined on a ring $\mathcal{R}$ is a measure if and only if

\[m\left( \lim\limits_{i \to \infty} E_i\right) = \lim\limits_{i\to \infty} m(E_i)\]

for every monotone increasing sequence $\{E_i\}$ of elements of $\mathcal{R}$ such that $\bigcup_{i=1}^\infty E_i \in \mathcal{R}$.

Let $(X_1, \mathcal{S}_1, \mu_1)$ be a measure space, let $(X_2, \mathcal{S}_2)$ be a measurable space and let $T$ be a measurable mapping from $X_1$ into $X_2$, i.e.

\[T^{-1}(E) = \{x\in X_1: Tx\in E\} \in \mathcal{S}_1\]

for all $E\in \mathcal{S}_2$. The measure generated by the mapping $T$ (denoted here by $\mu T^{-1}$) is the measure on $\mathcal{S}_2$ defined by

\[\mu T^{-1}(E) = \mu(T^{-1}E).\]

Let $(X, \mathcal{S}, \mu)$ be a measure space and let $X_1 \subset X$. Define $\mu_{X_1}$ on the sets $E$ from the $\sigma$-ring $ \mathcal{S} \cap X_1 = \{ E \cap X_1: E \in \mathcal{S}_1\}$ by

\[\mu_{X_1}(E) = \inf\limits_{E\subset F \in \mathcal{S}} \mu(F).\]

Then $(X_1, \mathcal{S} \cap X_1, \mu_{X_1})$ is a measure space; $\mu_{X_1}$ is called the restriction of the measure $\mu$ to $X_1$.

An atom of the space $(X, \mathcal{S}, \mu)$ (or of the measure $\mu$) is any set $E \in \mathcal{S}$ of positive measure such that if $F \subset E$, $F \in \mathcal{S}$, then either $\mu(F)=0$ or $\mu(F)=\mu(E)$. A measure space without atoms is called non-atomic or continuous (in this case $\mu$ is also called non-atomic or continuous). If $(X, \mathcal{S}, \mu)$ is a space with a non-atomic $\sigma$-finite measure and $E_1\in \mathcal{S}$, then for every $\alpha$ with $0 \le \alpha \le \mu(E_1)$ (possibly $\alpha = \infty$) there is an element $E_2 \in \mathcal{S}$ such that $E_2 \subset E_1$ and $\mu(E_2)=\alpha$.

A measure space $(X, \mathcal{S}, \mu_1)$ (or the measure $\mu$) is said to be complete if $E \in \mathcal{S}$, $F \subset E$, $\mu(E) = 0$ imply $F \in \mathcal{S}$. Every measure space $(X, \mathcal{S}, \mu)$ can be completed by adjoining to $\mathcal{S}$ all the sets of the form $E \cup N$ with $E \in \mathcal{S}$, $N \subset N'$, $N' \in S$, $\mu(N')=0$, and putting for such sets $\overline{\mu}(E\cup N) = \mu(E)$. The class of sets of the indicated form is a $\sigma$-ring, and $\overline{\mu}$ is a complete measure on it. The sets of null measure are called null sets. If the set of points of $X$ at which a property $Q$ is not satisfied is a null set, then property $Q$ is said to hold [[Almost- everywhere|almost-everywhere]].

Extension of measures.

A measure $ \mu _{2} $ is an extension of a measure $ \mu _{1} $ if $ \mu _{2} $ is an extension of $ \mu _{1} $ in the class of finitely-additive measures (see above). Every measure defined on a semi-ring $ {\mathcal P} $ admits a unique extension to a measure on the ring $ {\mathcal R}( {\mathcal P}) $ generated by $ {\mathcal P} $( the extension is realized in the same way as in the case of finitely-additive measures). Further, every measure $ \mu $ defined on a ring $ {\mathcal R} $ can be extended to a measure $ \mu^ \prime $ on the $ \sigma $- ring $ {\mathcal S} ( {\mathcal R}) $ generated by $ {\mathcal R} $; if $ \mu $ is $ \sigma $- finite, then $ \mu^ \prime $ is unique and $ \sigma $- finite. The value of $ \mu^ \prime $ on any set $ E \in {\mathcal S} ( {\mathcal R}) $ can be given by the formula

$$ \tag{*} \mu^ \prime (E) \ = \ \mathop{\rm inf}\nolimits \left \{ {\sum _ { i=1 } ^ \infty \mu (E _{i} )} : {E _{i} \in {\mathcal R} ,\ i = 1,\ 2 \dots \ E \subset \cup _ { i=1 } ^ \infty E _ i} \right \} . $$

A class of subsets of $ X $ is called hereditary if it contains, together with any set in the class, all its subsets. An outer measure is a set function $ m^ \star $, defined on a hereditary $ \sigma $- ring $ {\mathcal H} $( i.e. a class of sets which is simultaneously hereditary and a $ \sigma $- ring), which has the following properties:

1) $ 0 \leq m^ \star (E) \leq \infty $, $ m^ \star (\emptyset) = 0 $;

2) $ E \subset F $ implies $ m^ \star (E) \leq m^ \star (F \ ) $;

3) $ m^ \star ( \cup _ i=1^ \infty E _{i} ) \leq \sum _ i=1^ \infty m^ \star (E _{i} ) $.

Given a measure $ \mu $ on the ring $ {\mathcal R} $ one can construct an outer measure $ \mu^ \star $ on the hereditary $ \sigma $- ring $ {\mathcal H} ( {\mathcal R}) $ generated by $ {\mathcal R} $( $ {\mathcal H}( {\mathcal R}) $ consists of all sets that can be covered by a countable union of elements of $ {\mathcal R} $) by means of the formula

$$ \mu^ \star (E) \ = \ \mathop{\rm inf}\nolimits \left \{ {\sum _ { i=1 } ^ \infty \mu (E _{i} )} : {E _{i} \in {\mathcal R} ,\ i = 1,\ 2 \dots \ E \subset \cup _ { i=1 } ^ \infty E _ i} \right \} . $$

The outer measure $ \mu^ \star $ is called the outer measure induced by the measure $ \mu $.

Let $ m^ \star $ be an outer measure on a hereditary $ \sigma $- ring $ {\mathcal H} $ of subsets of $ X $. A set $ E \in {\mathcal H} $ is called $ m^ \star $- measurable if

$$ m^ \star (A) \ = \ m^ \star (A \cap E) + m^ \star (A \cap (X \setminus E)) $$

for every $ A \in {\mathcal H} $. The collection $ \overline{ {\mathcal S} }\; $ of $ m^ \star $- measurable sets is a $ \sigma $- ring which contains all sets of null outer measure. The set function $ \overline{m}\; $ on $ \overline{ {\mathcal S} }\; $ defined by the equality $ \overline{m}\; (E) = m^ \star (E) $ is a complete measure and is called the measure induced by the outer measure $ m^ \star $.

Suppose that $ \mu $ is a measure on a ring $ {\mathcal R} $ and that $ \mu^ \star $ is the outer measure on $ {\mathcal H} ( {\mathcal R}) $ induced by $ \mu $. Let $ \overline{ {\mathcal S} }\; $ and $ \overline \mu \; $ denote the collection of $ \mu^ \star $- measurable sets and the measure on $ \overline{ {\mathcal S} }\; $ induced by $ \mu^ \star $, respectively. Then $ \overline \mu \; $ is an extension of $ \mu $, and since $ {\mathcal S} ( {\mathcal R}) \subset \overline{ {\mathcal S} }\; $ it follows that the function $ \mu^ \prime $ on $ {\mathcal S}( {\mathcal R}) $ given by formula (*) is also a measure extending $ \mu $. If the original measure $ \mu $ on $ {\mathcal R} $ is $ \sigma $- finite, then the space $ (X,\ \overline{ {\mathcal S} }\; ,\ \overline \mu \; ) $ is the completion of the space $ (X,\ {\mathcal S}( {\mathcal R}) ,\ \mu^ \prime ) $( see (*)). If $ \mu $ is given on the $ \sigma $- ring $ {\mathcal S} $, then the induced outer measure $ \mu^ \star $ on the hereditary $ \sigma $- ring $ {\mathcal H}( {\mathcal S}) $ generated by $ {\mathcal S} $ is given by the formula

$$ \mu^ \star (E) \ = \ \mathop{\rm inf}\nolimits \{ {\mu (F \ )} : {E \subset F,\ F \in {\mathcal S}} \} . $$

Alongside with the outer measure $ \mu^ \star $, one defines the inner measure induced by the measure $ \mu $ on $ {\mathcal S} $. It is defined as

$$ \mu _ \star (E) \ = \ \sup \{ {\mu (F \ )} : {E \supset F,\ F \in {\mathcal S}} \} ,\ \ E \in {\mathcal H}( {\mathcal S}). $$

For every set $ E \in {\mathcal H}( {\mathcal S}) $ a measurable kernel $ E^ \prime $ and a measurable envelope $ E^{\prime\prime} $ are defined as elements of $ {\mathcal S} $ such that $ E^ \prime \subset E \subset E^{\prime\prime} $ and $ \mu (F ^ {\ \prime} ) = \mu (F ^ {\ \prime\prime} ) = 0 $ for all $ F ^ {\ \prime} ,\ F ^ {\ \prime\prime} \in {\mathcal S} $ such that $ F ^ {\ \prime} \subset E\setminus E^ \prime $, $ F ^ {\ \prime\prime} \subset E^{\prime\prime} \setminus E $. A measurable kernel exists always, while a measurable envelope exists whenever $ E $ has $ \sigma $- finite outer measure; moreover, $ \mu _ \star (E) = \mu (E^ \prime ) $ and $ \mu^ \star (E) = \mu (E^{\prime\prime} ) $. Let $ \mu $ be a measure on a ring $ {\mathcal R} $ and let $ \mu^ \prime $ be its extension to the $ \sigma $- ring $ {\mathcal S}( {\mathcal R}) $ generated by $ {\mathcal R} $. The inner measure $ \mu _ \star^ \prime $ on the subsets $ E $ of finite $ \mu $- measure can be expressed in terms of the outer measure $ \mu^ \star $( and hence $ \mu $):

$$ \mu _ \star^ \prime (A) \ = \ \mu (E) - \mu^ \star (E \setminus A),\ \ A \subset E. $$

Furthermore, a set $ F $ belonging to the hereditary $ \sigma $- ring $ {\mathcal H}( {\mathcal R}) $ with finite outer $ \mu^ \star $- measure is $ \mu^ \star $- measurable if and only if $ \mu^ \star (F \ ) = \mu _ \star^ \prime (F \ ) $. In case the original measure $ \mu $ on $ {\mathcal R} $ is totally finite, one has the following necessary and sufficient condition for the $ \mu^ \star $- measurability of a set $ E \subset X $:

$$ \mu (X) \ = \ \mu^ \star (E) + \mu^ \star (X\setminus E). $$

For totally-finite measures on $ {\mathcal R} $ this condition is frequently taken as the definition of $ \mu^ \star $- measurability of the set $ E $.

If $ (X,\ {\mathcal S} ,\ \mu ) $ is a space with a $ \sigma $- finite measure and $ X _{1} \dots X _{n} $ is a finite collection of elements of the hereditary $ \sigma $- ring $ {\mathcal H}( {\mathcal S}) $ generated by $ {\mathcal S} $, then on the $ \sigma $- ring $ \widetilde{ {\mathcal S} } $ generated by $ {\mathcal S} $ and the sets $ X _{1} \dots X _{n} $ one can define a measure $ \widetilde \mu $ which agrees with $ \mu $ on $ {\mathcal S} $.

Jordan, Lebesgue and Lebesgue–Stieltjes measures.

An example of an extension of a measure is provided by the Lebesgue measure in $ \mathbf R^{k} $. The intervals of the form

$$ I \ = \ \{ {(x _{1} \dots x _{k} )} : {a _{i} \leq x _{i} < b _{i} ,\ i = 1 \dots k} \} $$

form a semi-ring $ {\mathcal P} $ in $ \mathbf R^{k} $. For each such interval, let

$$ \lambda (I) \ = \ \prod _ { i=1 } ^ k (b _{i} - a _{i} ) $$

( $ \lambda (I) $ coincides with the volume of $ I $). The function $ \lambda $ is $ \sigma $- finite and countably additive on $ {\mathcal P} $ and admits a unique extension to a measure $ \lambda^ \prime $ on the $ \sigma $- ring $ {\mathcal S} $ generated by $ {\mathcal P} $; $ {\mathcal S} $ is identical with the $ \sigma $- ring of Borel sets (cf. Borel set) (or Borel-measurable sets) in $ \mathbf R^{k} $. The measure $ \lambda^ \prime $ was first defined by E. Borel in 1898 (see Borel measure). The completion $ \overline \lambda \; $ of $ \lambda^ \prime $( defined on $ \overline{ {\mathcal S} }\; \ $) is called the Lebesgue measure, and was introduced by H. Lebesgue in 1902 (see Lebesgue measure). A set belonging to the domain $ \overline{ {\mathcal S} }\; $ of $ \overline \lambda \; $ is called Lebesgue measurable. A bounded set $ E \subset \mathbf R^{k} $ belongs to $ \overline{ {\mathcal S} }\; $ if and only if $ \lambda (I) = \lambda^ \star (E) + \lambda^ \star (I\setminus E) $, where $ I \in {\mathcal P} $ is some interval containing $ E $; in this case $ \overline \lambda \; (E) = \lambda^ \star (E) $. A set $ E \subset \mathbf R^{k} $ belongs to $ \overline{ {\mathcal S} }\; $ if and only if for some sequence $ \{ r _{n} \} $, $ r _{n} > 0 $, $ n = 1,\ 2 \dots $ such that $ r _{n} \rightarrow \infty $, one has $ E \cap B _{ {r _ n}} \in \overline{ {\mathcal S} }\; $ for all $ n $, where $ B _{r} = \{ {x \in \mathbf R ^ k} : {\| x \| \leq r} \} $. The cardinality of the family of all Borel sets in $ \mathbf R^{k} $ is $ \mathfrak c $( the cardinality of the continuum), whereas the cardinality of the family of all Lebesgue-measurable sets is $ 2^{\mathfrak c} $, so that the inclusion $ {\mathcal S} \subset \overline{ {\mathcal S} }\; $ is strict, i.e. there exist Lebesgue-measurable sets that are not Borel measurable.

The Lebesgue measure $ \overline \lambda \; $ is invariant under linear orthogonal transformations $ A $ of $ \mathbf R^{k} $ as well as under translations by elements $ x \in \mathbf R^{k} $, i.e. $ \overline \lambda \; ( A E + x) = \overline \lambda \; (E) $ for all $ E \in {\mathcal S} $.

Using the axiom of choice one can show that there exist sets which are not Lebesgue measurable. On the straight line, for example, such a set can be obtained by picking one point in each coset in $ \mathbf R $ of the additive subgroup of rational numbers (Vitali's example).

Historically the Borel and Lebesgue measures in $ \mathbf R^{k} $ were preceded by the measure defined by C. Jordan in 1892 (see Jordan measure). The idea of the definition of the Jordan measure is very close to that of the classic definition of area and volume, which goes back to ancient Greece. Thus, a set $ E \subset \mathbf R^{k} $ is called Jordan measurable if there exist two sets, representable as finite unions of disjoint rectangles, one contained in $ E $ and the other containing $ E $, such that the difference of their volumes (defined in an obvious manner) is arbitrarily small. The Jordan measure of such a set is the infimum of the volumes of finite unions of rectangles covering $ E $. A Jordan-measurable set is also Lebesgue measurable, and its Jordan and Lebesgue measures are equal. The domain of the Jordan measure is merely a ring, and not a $ \sigma $- ring, which restricts considerably its domain of applicability.

The Lebesgue measure is a particular case of the more general Lebesgue–Stieltjes measure. The latter is defined by means of a real-valued function $ F $ on $ \mathbf R^{k} $ with the properties:

1) $ - \infty < F < \infty $;

2) $ \Delta _{ {b _{1} - a _ 1}} \dots \Delta _{ {b _{k} - a _ k}} F(a _{1} \dots a _{k} ) \geq 0 $ for $ a _{i} < b _{i} $, $ i = 1 \dots k $, where $ \Delta _{ {b _{i} - a _ i}} $ is the difference operator with step $ b _{i} - a _{i} $ taken at the point $ a _{i} $ with respect to the $ i $- th coordinate;

3) $ F(a _{1} \dots a _{k} ) \uparrow F(b _{1} \dots b _{k} ) $ as $ a _{i} \uparrow b _{i} $, $ i = 1 \dots k $.

Given such a function $ F $, the measure $ \mu _{F} $ of the interval

$$ I \ = \ \{ {(x _{1} \dots x _{k} )} : {a _{i} \leq x _{i} < b _{i} ,\ i = 1 \dots k} \} $$

is defined by the formula

$$ \mu _{F} (I) \ = \ \Delta _{ {b _{1} - a _ 1}} \dots \Delta _{ {b _{k} - a _ k}} F(a _{1} \dots a _{k} ). $$

It turns out that $ \mu _{F} $ is countably additive on the semi-ring of all such intervals and that it admits an extension to the $ \sigma $- algebra of Borel sets; the completion of this extension yields what is called the Lebesgue–Stieltjes measure corresponding to $ F $. For the particular choice

$$ F(x _{1} \dots x _{k} ) \ = \ x _{1} \dots x _{k} $$

one obtains the Lebesgue measure.

Measures in product spaces.

By definition, the product of two measurable spaces $ (X _{1} ,\ {\mathcal S} _{1} ) $, $ (X _{2} ,\ {\mathcal S} _{2} ) $ is the measurable space consisting of the set $ X _{1} \times X _{2} = \{ {(x _{1} ,\ x _{2} )} : {x _{1} \in X _{1} ,\ x _{2} \in X _ 2} \} $( the product of $ X _{1} $ and $ X _{2} $) and the $ \sigma $- ring $ {\mathcal S} _{1} \times {\mathcal S} _{2} $ of subsets of $ X $( the product of the $ \sigma $- rings $ {\mathcal S} _{1} $ and $ {\mathcal S} _{2} $) generated by the semi-ring $ {\mathcal P} $ of sets of the form

$$ E _{1} \times E _{2} \ = \ \{ {(x _{1} ,\ x _{2} )} : {x _{1} \in E _{1} ,\ x _{2} \in E _ 2} \} , $$

where $ E _{1} ,\ E _{2} \in {\mathcal S} $. If $ (X _{1} ,\ {\mathcal S} _{1} ,\ \mu _{1} ) $ and $ (X _{2} ,\ {\mathcal S} _{2} ,\ \mu _{s} ) $ are measure spaces, the formula

$$ \mu (E _{1} \times E _{2} ) \ = \ \mu _{1} (E _{1} ) \mu _{2} (E _{2} ),\ \ E _{1} \in {\mathcal S} _{1} ,\ \ E _{2} \in {\mathcal S} _{2} , $$

defines a measure on $ {\mathcal P} $; if $ \mu _{1} $ and $ \mu _{2} $ are $ \sigma $- finite, $ \mu $ extends uniquely to a measure on $ {\mathcal S} _{1} \times {\mathcal S} _{2} $, denoted by $ \mu _{1} \times \mu _{2} $. The measure $ \mu _{1} \times \mu _{2} $ and the space $ (X _{1} \times X _{2} ,\ {\mathcal S} _{1} \times {\mathcal S} _{2} ,\ \mu _{1} \times \mu _{2} ) $ are called, respectively, the product of the measures $ \mu _{1} $ and $ \mu _{2} $, and the product of the measure spaces $ (X _{1} ,\ {\mathcal S} _{1} ,\ \mu _{1} ) $ and $ (X _{2} ,\ {\mathcal S} _{2} ,\ \mu _{2} ) $. The completion of the product of the Lebesgue measure in $ \mathbf R^{k} $ and the Lebesgue measure in $ \mathbf R^{l} $ is the Lebesgue measure in $ \mathbf R^{k+l} $. Analogously one defines the product of an arbitrary finite number of measure spaces.

Let $ (X _{i} ,\ {\mathcal S} _{i} ,\ \mu _{i} ) $, $ i \in I $, be an arbitrary family of measure spaces such that $ \mu _{i} (X _{i} ) = 1 $, $ i \in I $. The product space $ X = \prod _{ {i \in I}} X _{i} $ is, by definition, the set of all functions on $ I $ such that the value at each $ i \in I $ is an element $ x _{i} \in X _{i} $. A measurable rectangle in $ X $ is any set of the form $ \prod _{ {i \in I}} E _{i} $, where $ E _{i} \in {\mathcal S} _{i} $ and only finitely many sets $ E _{i} $ are different from $ X _{i} $. The family of measurable rectangles forms a semi-ring $ {\mathcal P} $. The $ \sigma $- ring generated by $ {\mathcal P} $ is denoted by $ \prod _{ {i \in I}} {\mathcal S} _{i} $ and is called the product of the $ \sigma $- rings $ {\mathcal S} _{i} $. Now, let $ \mu $ be the function on $ {\mathcal P} $ defined by $ \mu (E) = \prod _{ {i \in I}} \mu _{i} (E _{i} ) $ for $ E = \prod _{ {i \in I}} E _{i} $. The function $ \mu $ thus defined is a measure which admits a unique extension to a measure on $ \prod _{ {i \in I}} {\mathcal S} _{i} $, denoted by $ \prod _{ {i \in I}} \mu _{i} $. The measure space $ ( \prod _{ {i \in I}} X _{i} ,\ \prod _{ {i \in I}} {\mathcal S} _{i} ,\ \prod _{ {i \in I}} \mu _{i} ) $ is called the product of the spaces $ (X _{i} ,\ {\mathcal S} _{i} ,\ \mu _{i} ) $, $ i \in I $.

The product of an arbitrary number of measure spaces is a particular case of the following, more general, scheme, which plays an important role in probability theory. Let $ (X _{i} ,\ {\mathcal S} _{i} ) $, $ i \in I $, be a family of measurable spaces (each $ {\mathcal S} _{i} $ is a $ \sigma $- algebra), and suppose that for each finite subset $ I _{1} \subset I $ there is given a probability measure $ \mu _{ {I _ 1}} $ on the measurable spaces $ ( \prod _{ {i \in I _ 1}} X _{i} ,\ \prod _{ {i \in I _ 1}} \in {\mathcal S} _{i} ) $( the product of measures corresponds to the case that $ \mu _{ {I _ 1}} = \prod _{ {i \in I _ 1}} \mu _{i} $ for all finite $ I _{1} \subset I $). Suppose further that each two measures $ \mu _{ {I _ 1}} ,\ \mu _{ {I _ 2}} $ are compatible in the sense that if $ I _{1} \subset I _{2} $ and $ p _{21} $ is the projection of $ \prod _{ {i \in I _ 2}} X _{i} $ onto $ \prod _{ {i \in I _ 1}} X _{i} $, then $ \mu _{ {I _ 1}} (E) = \mu _{ {I _ 2}} p _ 21^{-1} (E) $ for all $ E \in \prod _{ {i \in I _ 1}} {\mathcal S} _{i} $( by definition, $ p _{21} $ is the mapping of $ \prod _{ {i \in I _ 2}} X _{i} $ onto $ \prod _{ {i \in I _ 1}} X _{i} $ such that $ (p _{21} (x )) _{i} = x _{i} $ for all $ i \in I _{1} $). The following question arises: Is there a probability measure on $ \prod _{ {i \in I}} {\mathcal S} _{i} $ such that $ \mu _{ {I _ 1}} (E) = \mu p^{-1} (E) $ for every finite $ I _{1} \subset I $ and every $ E \in \prod _{ {i \in I _ 1}} {\mathcal S} _{i} $, where $ p $ denotes the projection of $ \prod _{ {i \in I}} X _{i} $ onto $ \prod _{ {i \in I _ 1}} X _{i} $? It turns out that such a measure does not always exist, and additional conditions must be imposed to guarantee its existence. One such condition is perfectness of the measures $ \mu _{i} $( corresponding to the one-point sets $ i \in I $). The notion of a perfect measure was first introduced by B.V. Gnedenko and A.N. Kolmogorov [6]. A space $ (X,\ {\mathcal S} ,\ \mu ) $ with a totally-finite measure, as well as the measure $ \mu $ itself, is called perfect if for every $ {\mathcal S} $- measurable real-valued function $ f $ on $ X $ there is a Borel set $ B \subset f(X) $ such that $ \mu (f ^ {\ -1} (B)) = \mu (X) $. The perfectness assumption eliminates a series of "pathological" phenomena that arise in general measure theory.

Measures in topological spaces.

The study of measures in topological spaces is usually concerned with measures defined on sets connected in some way or another with the topology of the underlying space. One of the typical approaches is the following. Let $ X $ be an arbitrary topological space and let $ {\mathcal Z} $ be the class of subsets of the form $ f ^ {\ -1} (F \ ) $, where $ f $ is a continuous real-valued function on $ X $ and $ F \subset \mathbf R^{1} $ is a closed set. Let $ \mathfrak A $ be the algebra generated by the class $ {\mathcal Z} $ and let $ {\mathcal B} $ be the $ \sigma $- algebra generated by $ {\mathcal Z} $( $ {\mathcal B} $ is called the $ \sigma $- algebra of Baire sets, cf. also Algebra of sets). Now let $ {\mathcal M} $ be the class of totally-finite finitely-additive measures $ m $ on $ \mathfrak A $ that are regular in the sense that

$$ m(E) \ = \ \sup \{ {m(Z)} : {Z \subset E,\ Z \in {\mathcal Z}} \} $$

for all $ E \in \mathfrak A $. In $ {\mathcal M} $ one distinguishes the subclasses $ {\mathcal M} _ \sigma $, $ {\mathcal M} _ \tau $ and $ {\mathcal M} _{t} $ formed by the (finitely-additive) measures possessing additional smoothness properties. By definition, $ \mu \in {\mathcal M} _ \sigma $ if $ \mu (Z _{n} ) \downarrow 0 $ for every sequence $ Z _{n} \downarrow \emptyset $, $ Z _{n} \in {\mathcal Z} $( this property is equivalent to the countable additivity of $ \mu $; the measures from $ {\mathcal M} _ \sigma $ admit unique extensions to $ {\mathcal B} $ and hereafter it is assumed that they are given on $ {\mathcal B} $); $ \mu \in {\mathcal M} _ \tau $ if $ \mu ( {\mathcal Z} _ \alpha ) \downarrow 0 $ for every net $ Z _ \alpha \downarrow \emptyset $, $ Z _ \alpha \in {\mathcal Z} $; and $ \mu \in {\mathcal M} _{t} $ if for every $ \epsilon > 0 $ there is a compact set $ K $ such that $ \mu (E) < \epsilon $ whenever $ E \subset X\setminus K $, $ E \in \mathfrak A $.

The inclusions $ {\mathcal M} \supset {\mathcal M} _ \sigma \supset {\mathcal M} _ \tau \supset {\mathcal M} _{t} $ hold. The elements of $ {\mathcal M} _ \sigma $ are called Baire measures.

There is an intimate connection between the measures belonging to $ {\mathcal M} $ and the linear functionals on the space $ C(X) $ of bounded continuous functions on $ X $. Namely, the formula

$$ \Lambda (f \ ) \ = \ \int\limits _ { X } f \ dm $$

establishes a one-to-one correspondence between the finitely-additive measures $ m \in {\mathcal M} $ and the non-negative linear functionals $ \Lambda $ on $ C(X) $( non-negative means that $ \Lambda (f \ ) \geq 0 $ whenever $ f(x) \geq 0 $, $ x \in X $). Moreover, for every set $ Z \in {\mathcal Z} $,

$$ m(Z) \ = \ \mathop{\rm inf}\nolimits \{ {\Lambda (f \ )} : {\chi _{Z} \leq f \leq 1 } \} , $$

where $ \chi _{Z} $ is the indicator function of $ Z $. This correspondence takes the measures from $ {\mathcal M} _ \sigma $ into $ \sigma $- smooth functionals $ \Lambda $( i.e. functionals $ \Lambda $ with the property that $ \Lambda (f _{n} ) \rightarrow 0 $ if $ f _{n} \downarrow 0 $ in $ C(X) $), the measures from $ {\mathcal M} _ \tau $ into $ \tau $- smooth functionals $ \Lambda $( i.e. functionals such that $ \Lambda (f _ \alpha ) \rightarrow 0 $ for every net $ f _ \alpha \downarrow 0 $ in $ C(X) $), and the measures from $ {\mathcal M} _{t} $ into dense functionals $ \Lambda $( i.e. with the property that $ \Lambda (f _ \alpha ) \rightarrow 0 $ for every net $ f _ \alpha $ in $ C(X) $ such that $ \| f _ \alpha \| \leq 1 $ for all $ \alpha $ and $ f _ \alpha \rightarrow 0 $ uniformly on compact subsets; here $ \| \cdot \| $ is the uniform norm).

The space $ {\mathcal M} $ is usually endowed with the weak topology $ w $, in which a basis of neighbourhoods consists of the sets of the form

$$ U(m _{0} ; \ f _{1} \dots f _{n} ,\ \epsilon )\ = $$

$$ = \ \left \{ m : \ \left | \int\limits _ { X } f _{k} \ (dm-dm _{0} ) \right | < \epsilon ,\ k = 1 \dots n,\ f _{1} \dots f _{n} \in C(X) \right \} . $$

With the topology $ w $, $ {\mathcal M} $ is a completely-regular Hausdorff space. Convergence in the topology $ w $ is usually denoted by the symbol $ \Rightarrow $. For the convergence of a net $ m _ \alpha $ to $ m $: $ m _ \alpha \Rightarrow m $, it is necessary and sufficient that $ m _ \alpha (X) \rightarrow m(X) $ and $ \lim\limits \ \sup \ m _ \alpha (Z) \leq m(Z) $ for all $ Z \in {\mathcal Z} $. Another necessary and sufficient condition for the convergence $ m _ \alpha \Rightarrow m $ is that $ m _ \alpha (E) \rightarrow m(E) $ for all $ E \in \mathfrak A $ such that there are $ Z _{1} ,\ Z _{2} \in {\mathcal Z} $ with $ X \setminus E \subset Z _{1} $, $ E \subset Z _{2} $, and $ m(Z _{1} \cap Z _{2} ) = 0 $. If the space $ X $ is completely regular and Hausdorff, then $ {\mathcal M} _ \tau $ is metrizable if and only if $ X $ is metrizable. If $ X $ is metrizable, then $ {\mathcal M} _ \tau $ admits a metric in which it is separable if and only if $ X $ is separable, and it admits a metric in which it is complete if and only if $ X $ has a complete metric. If $ X $ is metrizable, then $ {\mathcal M} _ \sigma $ is metrizable if and only if it is metrizable by the Lévy–Prokhorov metric.

The space $ {\mathcal M} _ \sigma $ is sequentially closed in $ {\mathcal M} $( Aleksandrov's theorem). A set $ A \subset {\mathcal M} $ is called tight if $ \sup \{ {m(X)} : {m \in A} \} < \infty $ and if for every $ \epsilon > 0 $ there is a compact set $ K $ such that $ m(E) < \epsilon $ for all $ E \subset X\setminus K $, $ m \in A $ and $ E \in \mathfrak A $. If $ A \subset {\mathcal M} _ \sigma $ is tight, then $ A $ is relatively compact in $ {\mathcal M} _ \sigma $; conversely, if $ X $ is metrizable and topologically complete, then $ A \subset {\mathcal M} _ \sigma $ is relatively compact, and if every measure in $ A $ is concentrated on some separable subset of $ X $, then $ A $ is tight (Prokhorov's theorem).

Under certain conditions the elements of $ {\mathcal M} _ \sigma $ can be extended to Borel measures, i.e. measures defined on the $ \sigma $- algebra of Borel sets (see Borel set; Borel measure). For example, if $ X $ is a normal countably-paracompact Hausdorff space, then every measure $ \mu \in {\mathcal M} _ \sigma $ admits a unique extension to a regular Borel measure. If $ X $ is completely regular and Hausdorff, then every $ \tau $- smooth (tight) Baire measure admits a unique extension to a $ \tau $- smooth (tight) Borel measure.

The support of a Baire (Borel) measure is the smallest set $ Z \in {\mathcal Z} $( respectively, the smallest closed set) the measure of which is equal to the measure of the whole space. Every $ \tau $- smooth measure has a support.

Often, when measures in topological spaces (especially in locally compact Hausdorff spaces) are considered, it is assumed that the Borel and Baire measures are given on less-wide classes of sets, more precisely — on $ \sigma $- rings generated by compact sets and, respectively, compact $ G _ \delta $- sets.

Let $ G $ be a locally compact Hausdorff topological group. A left Haar measure on $ G $ is a measure defined on the $ \sigma $- ring generated by all compact subsets that does not vanish identically and is such that $ \mu (xE) = \mu (E) $ for all $ x \in G $ and $ E $ in the domain of $ \mu $. A right Haar measure is defined in the same manner but with the condition $ \mu (xE) = \mu (E) $ replaced by $ \mu (Ex) = \mu (E) $. On any group of the type considered a left Haar measure exists and is unique (up to a multiplicative positive constant). Every left Haar measure is regular in the sense that $ \mu (E) = \sup \{ {\mu (K)} : {K \subset E} \} $, where $ K $ are compact sets. The right Haar measure has analogous properties. The Lebesgue measure on $ \mathbf R^{k} $ is a particular case of the Haar measure. See also Measure in a topological vector space.

Isomorphism of measure spaces.

Let $ (X,\ {\mathcal S} ,\ \mu ) $ be a measure space. Call two sets $ E,\ E^ \prime \in {\mathcal S} $ $ \mu $- equal (written $ E = E^ \prime $ $ [ \mu ] $) if $ \mu (E \Delta E^ \prime ) = 0 $( where $ E \Delta E^ \prime $ denotes the symmetric difference of $ E $ and $ E^ \prime $, cf. Symmetric difference of sets). Denote by $ {\mathcal S} _ \mu $ the class of sets $ {\mathcal S} $ with this equality relation. In $ {\mathcal S} _ \mu $ the set-theoretic operations, performed a finite (or countable) number of times are correctly defined: for example, if $ E _{1} = E _ 1^ \prime $ $ [ \mu ] $ and $ E _{2} = E _ 2^ \prime $ $ [ \mu ] $, then $ E _{1} \cup E _{2} = E _ 1^ \prime \cup E _ 2^ \prime $ $ [ \mu ] $. The measure $ \mu $ is carried over, in an obvious manner, to $ {\mathcal S} _ \mu $.

Let $ \widetilde{ {\mathcal S} } _ \mu $ be the subset of $ {\mathcal S} _ \mu $ consisting of the sets of finite measure. The function $ \rho (E,\ E^ \prime ) = \mu (E \Delta E^ \prime ) $ on $ \widetilde{ {\mathcal S} } _ \mu \times \widetilde{ {\mathcal S} } _ \mu $ is a metric. The measure space $ (X,\ {\mathcal S} ,\ \mu ) $ is said to be separable if the space $ \widetilde{ {\mathcal S} } _ \mu $ with metric $ \rho $ is separable. If $ (X,\ {\mathcal S} ,\ \mu ) $ is a space with a $ \sigma $- finite measure and the $ \sigma $- ring $ {\mathcal S} $ is countably generated (i.e. there is a countable family $ \{ E _{n} \} \subset {\mathcal S} $ such that $ {\mathcal S} $ is the smallest $ \sigma $- ring that contains this family), then the metric space $ \widetilde{ {\mathcal S} } _ \mu $ is separable.

Two measure spaces, $ (X _{1} ,\ {\mathcal S} _{1} ,\ \mu _{1} ) $ and $ (X _{2} ,\ {\mathcal S} _{2} ,\ \mu _{2} ) $ are said to be isomorphic if there is a one-to-one mapping $ \phi $ of $ ( {\mathcal S} _{1} ) _{ {\mu _ 1}} $ onto $ ( {\mathcal S} _{2} ) _{ {\mu _ 2}} $ such that

$$ \phi (E\setminus F \ ) \ = \ \phi (E) \setminus \phi (F \ ) ,\ \ \phi (E \cup F \ ) \ = \ \phi (E) \cup \phi (F \ ) $$

and

$$ \mu _{1} (E) \ = \ \mu _{2} ( \phi (E)) \ \ \textrm{ for \ all } \ E,\ F \in ( {\mathcal S} _{1} ) _{ {\mu _ 1}} . $$

Now, let $ (X,\ {\mathcal S} ,\ \mu ) $ be an arbitrary space with a totally-finite measure. There is a partition of $ X $ into disjoint sets $ X _{n} \in {\mathcal S} $, $ n = 1,\ 2 \dots $ such that the restriction of $ \mu $ to $ X _{n} $ is isomorphic either to a measure concentrated at one point or to a measure which is equal, up to a positive factor, to the direct product $ \prod _{ {i \in I}} (U _{i} ,\ {\mathcal U} _{i} ,\ u _{i} ) $, where $ U _{i} = \{ 0,\ 1 \} $, $ u _{i} ( \{ 0 \} ) = u _{i} ( \{ 1 \} ) = 1/2 $, and the set $ I $ may have arbitrary cardinality (the Maharan–Kolmogorov theorem). If $ (X,\ {\mathcal S} ,\ \mu ) $ is separable, non-atomic and $ \mu (X) = 1 $, then it is isomorphic to the space $ \prod _{ {i \in I}} (U _{i} ,\ {\mathcal U} _{i} ,\ u _{i} ) $ with $ I $ countable, which in turn is isomorphic to the unit interval with the Lebesgue measure.

Side by side with the theory of measures regarded as functions on subsets of some set, the theory of measures as functions on the elements of a Boolean ring (or on a Boolean algebra) has been developed; these theories are in many respects parallel. Another widespread construction of measures goes back to W. Young and P. Daniell (see [12]). Theories dealing with measures with real or complex values, or with values belonging to some algebraic structure, were developed in addition to the theory of positive measures.

References

[1] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05
[2] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[3] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[4] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
[5] J. Neveu, "Mathematical foundations of the calculus of probabilities" , Holden-Day (1965) (Translated from French) MR0198505
[6] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001
[7] V.S. Varadarajan, "Measures on topological spaces" Mat. Sb. , 55 : 1 (1961) pp. 35–100 (In Russian) MR0148838
[8] K.R. Parthasarathy, "Probability measures on metric spaces" , Acad. Press (1967) MR0226684 Zbl 0153.19101
[9] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[10] R. Sikorski, "Boolean algebras" , Springer (1969) MR0249336 MR0242724 Zbl 0191.31505
[11] D.A. Vladimirov, "Boolesche Algebren" , Akademie Verlag (1978) (Translated from Russian) MR0524392 Zbl 0385.06018
[12] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[13] J. Diestel, J.J. Uhl jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977) MR0453964 Zbl 0369.46039

Comments

Properties 1) and 2) listed under the heading "Properties of measure spaces" are usually called Fatou's lemma, cf. Fatou theorem.

The procedure for extending a measure, as described under the heading "Extension of measures" , is due to C. Carathéodory, and one often speaks of Carathéodory extension, with the accompanying phrases Carathéodory extension theorem and Carathéodory outer (inner) measure (cf. Carathéodory measure).

Recall that a ring (respectively, a $ \sigma $- ring) $ {\mathcal A} $ of subsets of a set $ X $ such that $ A \in {\mathcal A} $ implies $ X \setminus A \in {\mathcal A} $, is called a Boolean algebra or an algebra (respectively, a $ \sigma $- algebra or a $ \sigma $- field, cf. also Algebra of sets). Usually, in a measure space $ ( X ,\ {\mathcal S} ,\ \mu ) $ the $ \sigma $- ring $ {\mathcal S} $ can be proved to be a $ \sigma $- field (this holds, in particular, if $ \mu ( X ) < \infty $).

The phrase "totally (s-) finite" is seldom used.

Borel has given very nice ideas in order to construct the measure $ \lambda^ \prime $, but Lebesgue was the first to give a satisfactory construction of it, as a byproduct of the construction of $ \overline \lambda \; $.

A product space is also often written as a (kind of) tensor product: $ ( X _{1} \times X _{2} ,\ {\mathcal S} _{1} \otimes {\mathcal S} _{2} ,\ \mu _{1} \otimes \mu _{2} ) $.

A family of measurable spaces $ ( X _{i} ,\ {\mathcal S} _{i} ) _{i} $ with compatible probability measures on each finite product is called a projective system of measure spaces, and the corresponding probability measure on $ \prod {X _ i} $, if it exists, is called the projective limit; it exists if $ I $ is countable (the Ionescu–Tulcea theorem, cf. [5]).

Suppose that $ X $ is a topological space and $ {\mathcal S} $ is its Borel $ \sigma $- field; then $ ( X ,\ {\mathcal S} ,\ \mu ) $ is perfect for every finite measure $ \mu $ if $ X $ is a Polish space or, more generally, a Luzin space (in which case $ ( X ,\ {\mathcal S} ) $ is often called a standard measurable space) or, still more generally, a Suslin space (in which case $ ( X ,\ {\mathcal S} ) $ is sometimes called a Blackwell measurable space) (cf. (the editorial comments to) Descriptive set theory).

The converse part of Prokhorov's theorem is not true when $ X $ is the space of rational numbers, or, more generally, when $ X $ is a Luzin space which is not Polish. See [a1].

In the abstract setting, whenever $ ( \mu _{n} ) $ is a sequence of finite measures on $ ( X ,\ {\mathcal S} ) $, where $ {\mathcal S} $ is a $ \sigma $- field, such that

$$ m (A) \ = \ \lim\limits _ { n } \ \mu _{n} (A) $$

exists for any $ A \in {\mathcal S} $, then $ m $ is also a measure (the Vitali–Hahn–Saks theorem, cf. [3] or [5]).

References

[a1] D. Preiss, "Metric spaces in which Prokhorov's theorem is not valid" Z. Wahrscheinlichkeitstheor. Verw. Gebiete , 27 (1973) pp. 109–116
[a2] D. Cohn, "Measure theory" , Birkhäuser (1980) MR0578344 Zbl 0436.28001
[a3] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , I , Springer (1979) MR0551496 Zbl 0416.43001
[a4] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
How to Cite This Entry:
Measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure&oldid=44380
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article