Measure
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
- $ \emptyset \in \mathcal{P} $
- $ E_1, E_2 \in \mathcal{P} \implies E_1 \cap E_2 \in \mathcal{P} $
- $ 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
- $\emptyset \in \mathcal{R}$
- $E_1, E_2 \in \mathcal{R} \implies E_1 \cup E_2 \in \mathcal{R}, E_1 \setminus E_2 \in \mathcal{R}$.
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
- $\emptyset \in \mathcal{S} $
- $E_1, E_2 \in \mathcal{S} \implies E_1 \setminus E_2 \in \mathcal{S}$
- $E_i \in \mathcal{S} \quad (i = 1, 2, \dots) \implies \bigcup_{i=1}^\infty E_i \in S$.
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 be an arbitrary sequence of measurable sets. Then
1) ;
2) if for some
, then
![]() |
3) if exists and the condition in 2) is satisfied, then
![]() |
A finitely-additive measure defined on a ring
is a measure if and only if
![]() |
for every monotone increasing sequence of elements of
such that
.
Let be a measure space, let
be a measurable space and let
be a measurable mapping from
into
, i.e.
![]() |
for all . The measure generated by the mapping
(denoted here by
) is the measure on
defined by
![]() |
Let be a measure space and let
. Define
on the sets
from the
-ring
by
![]() |
Then is a measure space;
is called the restriction of the measure
to
.
An atom of the space (or of the measure
) is any set
of positive measure such that if
,
, then either
or
. A measure space without atoms is called non-atomic or continuous (in this case
is also called non-atomic or continuous). If
is a space with a non-atomic
-finite measure and
, then for every
with
(possibly
) there is an element
such that
and
.
A measure space (or the measure
) is said to be complete if
,
,
imply
. Every measure space
can be completed by adjoining to
all the sets of the form
with
,
,
,
, and putting for such sets
. The class of sets of the indicated form is a
-ring, and
is a complete measure on it. The sets of null measure are called null sets. If the set of points of
at which a property
is not satisfied is a null set, then property
is said to hold almost-everywhere.
Extension of measures.
A measure is an extension of a measure
if
is an extension of
in the class of finitely-additive measures (see above). Every measure defined on a semi-ring
admits a unique extension to a measure on the ring
generated by
(the extension is realized in the same way as in the case of finitely-additive measures). Further, every measure
defined on a ring
can be extended to a measure
on the
-ring
generated by
; if
is
-finite, then
is unique and
-finite. The value of
on any set
can be given by the formula
![]() | (*) |
A class of subsets of is called hereditary if it contains, together with any set in the class, all its subsets. An outer measure is a set function
, defined on a hereditary
-ring
(i.e. a class of sets which is simultaneously hereditary and a
-ring), which has the following properties:
1) ,
;
2) implies
;
3) .
Given a measure on the ring
one can construct an outer measure
on the hereditary
-ring
generated by
(
consists of all sets that can be covered by a countable union of elements of
) by means of the formula
![]() |
The outer measure is called the outer measure induced by the measure
.
Let be an outer measure on a hereditary
-ring
of subsets of
. A set
is called
-measurable if
![]() |
for every . The collection
of
-measurable sets is a
-ring which contains all sets of null outer measure. The set function
on
defined by the equality
is a complete measure and is called the measure induced by the outer measure
.
Suppose that is a measure on a ring
and that
is the outer measure on
induced by
. Let
and
denote the collection of
-measurable sets and the measure on
induced by
, respectively. Then
is an extension of
, and since
it follows that the function
on
given by formula (*) is also a measure extending
. If the original measure
on
is
-finite, then the space
is the completion of the space
(see (*)). If
is given on the
-ring
, then the induced outer measure
on the hereditary
-ring
generated by
is given by the formula
![]() |
Alongside with the outer measure , one defines the inner measure induced by the measure
on
. It is defined as
![]() |
For every set a measurable kernel
and a measurable envelope
are defined as elements of
such that
and
for all
such that
,
. A measurable kernel exists always, while a measurable envelope exists whenever
has
-finite outer measure; moreover,
and
. Let
be a measure on a ring
and let
be its extension to the
-ring
generated by
. The inner measure
on the subsets
of finite
-measure can be expressed in terms of the outer measure
(and hence
):
![]() |
Furthermore, a set belonging to the hereditary
-ring
with finite outer
-measure is
-measurable if and only if
. In case the original measure
on
is totally finite, one has the following necessary and sufficient condition for the
-measurability of a set
:
![]() |
For totally-finite measures on this condition is frequently taken as the definition of
-measurability of the set
.
If is a space with a
-finite measure and
is a finite collection of elements of the hereditary
-ring
generated by
, then on the
-ring
generated by
and the sets
one can define a measure
which agrees with
on
.
Jordan, Lebesgue and Lebesgue–Stieltjes measures.
An example of an extension of a measure is provided by the Lebesgue measure in . The intervals of the form
![]() |
form a semi-ring in
. For each such interval, let
![]() |
( coincides with the volume of
). The function
is
-finite and countably additive on
and admits a unique extension to a measure
on the
-ring
generated by
;
is identical with the
-ring of Borel sets (cf. Borel set) (or Borel-measurable sets) in
. The measure
was first defined by E. Borel in 1898 (see Borel measure). The completion
of
(defined on
) is called the Lebesgue measure, and was introduced by H. Lebesgue in 1902 (see Lebesgue measure). A set belonging to the domain
of
is called Lebesgue measurable. A bounded set
belongs to
if and only if
, where
is some interval containing
; in this case
. A set
belongs to
if and only if for some sequence
,
,
such that
, one has
for all
, where
. The cardinality of the family of all Borel sets in
is
(the cardinality of the continuum), whereas the cardinality of the family of all Lebesgue-measurable sets is
, so that the inclusion
is strict, i.e. there exist Lebesgue-measurable sets that are not Borel measurable.
The Lebesgue measure is invariant under linear orthogonal transformations
of
as well as under translations by elements
, i.e.
for all
.
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 of the additive subgroup of rational numbers (Vitali's example).
Historically the Borel and Lebesgue measures in 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
is called Jordan measurable if there exist two sets, representable as finite unions of disjoint rectangles, one contained in
and the other containing
, 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
. 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
-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 on
with the properties:
1) ;
2) for
,
, where
is the difference operator with step
taken at the point
with respect to the
-th coordinate;
3) as
,
.
Given such a function , the measure
of the interval
![]() |
is defined by the formula
![]() |
It turns out that is countably additive on the semi-ring of all such intervals and that it admits an extension to the
-algebra of Borel sets; the completion of this extension yields what is called the Lebesgue–Stieltjes measure corresponding to
. For the particular choice
![]() |
one obtains the Lebesgue measure.
Measures in product spaces.
By definition, the product of two measurable spaces ,
is the measurable space consisting of the set
(the product of
and
) and the
-ring
of subsets of
(the product of the
-rings
and
) generated by the semi-ring
of sets of the form
![]() |
where . If
and
are measure spaces, the formula
![]() |
defines a measure on ; if
and
are
-finite,
extends uniquely to a measure on
, denoted by
. The measure
and the space
are called, respectively, the product of the measures
and
, and the product of the measure spaces
and
. The completion of the product of the Lebesgue measure in
and the Lebesgue measure in
is the Lebesgue measure in
. Analogously one defines the product of an arbitrary finite number of measure spaces.
Let ,
, be an arbitrary family of measure spaces such that
,
. The product space
is, by definition, the set of all functions on
such that the value at each
is an element
. A measurable rectangle in
is any set of the form
, where
and only finitely many sets
are different from
. The family of measurable rectangles forms a semi-ring
. The
-ring generated by
is denoted by
and is called the product of the
-rings
. Now, let
be the function on
defined by
for
. The function
thus defined is a measure which admits a unique extension to a measure on
, denoted by
. The measure space
is called the product of the spaces
,
.
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 ,
, be a family of measurable spaces (each
is a
-algebra), and suppose that for each finite subset
there is given a probability measure
on the measurable spaces
(the product of measures corresponds to the case that
for all finite
). Suppose further that each two measures
are compatible in the sense that if
and
is the projection of
onto
, then
for all
(by definition,
is the mapping of
onto
such that
for all
). The following question arises: Is there a probability measure on
such that
for every finite
and every
, where
denotes the projection of
onto
? 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
(corresponding to the one-point sets
). The notion of a perfect measure was first introduced by B.V. Gnedenko and A.N. Kolmogorov [6]. A space
with a totally-finite measure, as well as the measure
itself, is called perfect if for every
-measurable real-valued function
on
there is a Borel set
such that
. 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 be an arbitrary topological space and let
be the class of subsets of the form
, where
is a continuous real-valued function on
and
is a closed set. Let
be the algebra generated by the class
and let
be the
-algebra generated by
(
is called the
-algebra of Baire sets, cf. also Algebra of sets). Now let
be the class of totally-finite finitely-additive measures
on
that are regular in the sense that
![]() |
for all . In
one distinguishes the subclasses
,
and
formed by the (finitely-additive) measures possessing additional smoothness properties. By definition,
if
for every sequence
,
(this property is equivalent to the countable additivity of
; the measures from
admit unique extensions to
and hereafter it is assumed that they are given on
);
if
for every net
,
; and
if for every
there is a compact set
such that
whenever
,
.
The inclusions hold. The elements of
are called Baire measures.
There is an intimate connection between the measures belonging to and the linear functionals on the space
of bounded continuous functions on
. Namely, the formula
![]() |
establishes a one-to-one correspondence between the finitely-additive measures and the non-negative linear functionals
on
(non-negative means that
whenever
,
). Moreover, for every set
,
![]() |
where is the indicator function of
. This correspondence takes the measures from
into
-smooth functionals
(i.e. functionals
with the property that
if
in
), the measures from
into
-smooth functionals
(i.e. functionals such that
for every net
in
), and the measures from
into dense functionals
(i.e. with the property that
for every net
in
such that
for all
and
uniformly on compact subsets; here
is the uniform norm).
The space is usually endowed with the weak topology
, in which a basis of neighbourhoods consists of the sets of the form
![]() |
![]() |
With the topology ,
is a completely-regular Hausdorff space. Convergence in the topology
is usually denoted by the symbol
. For the convergence of a net
to
:
, it is necessary and sufficient that
and
for all
. Another necessary and sufficient condition for the convergence
is that
for all
such that there are
with
,
, and
. If the space
is completely regular and Hausdorff, then
is metrizable if and only if
is metrizable. If
is metrizable, then
admits a metric in which it is separable if and only if
is separable, and it admits a metric in which it is complete if and only if
has a complete metric. If
is metrizable, then
is metrizable if and only if it is metrizable by the Lévy–Prokhorov metric.
The space is sequentially closed in
(Aleksandrov's theorem). A set
is called tight if
and if for every
there is a compact set
such that
for all
,
and
. If
is tight, then
is relatively compact in
; conversely, if
is metrizable and topologically complete, then
is relatively compact, and if every measure in
is concentrated on some separable subset of
, then
is tight (Prokhorov's theorem).
Under certain conditions the elements of can be extended to Borel measures, i.e. measures defined on the
-algebra of Borel sets (see Borel set; Borel measure). For example, if
is a normal countably-paracompact Hausdorff space, then every measure
admits a unique extension to a regular Borel measure. If
is completely regular and Hausdorff, then every
-smooth (tight) Baire measure admits a unique extension to a
-smooth (tight) Borel measure.
The support of a Baire (Borel) measure is the smallest set (respectively, the smallest closed set) the measure of which is equal to the measure of the whole space. Every
-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 -rings generated by compact sets and, respectively, compact
-sets.
Let be a locally compact Hausdorff topological group. A left Haar measure on
is a measure defined on the
-ring generated by all compact subsets that does not vanish identically and is such that
for all
and
in the domain of
. A right Haar measure is defined in the same manner but with the condition
replaced by
. 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
, where
are compact sets. The right Haar measure has analogous properties. The Lebesgue measure on
is a particular case of the Haar measure. See also Measure in a topological vector space.
Isomorphism of measure spaces.
Let be a measure space. Call two sets
-equal (written
) if
(where
denotes the symmetric difference of
and
, cf. Symmetric difference of sets). Denote by
the class of sets
with this equality relation. In
the set-theoretic operations, performed a finite (or countable) number of times are correctly defined: for example, if
and
, then
. The measure
is carried over, in an obvious manner, to
.
Let be the subset of
consisting of the sets of finite measure. The function
on
is a metric. The measure space
is said to be separable if the space
with metric
is separable. If
is a space with a
-finite measure and the
-ring
is countably generated (i.e. there is a countable family
such that
is the smallest
-ring that contains this family), then the metric space
is separable.
Two measure spaces, and
are said to be isomorphic if there is a one-to-one mapping
of
onto
such that
![]() |
and
![]() |
Now, let be an arbitrary space with a totally-finite measure. There is a partition of
into disjoint sets
,
such that the restriction of
to
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
, where
,
, and the set
may have arbitrary cardinality (the Maharan–Kolmogorov theorem). If
is separable, non-atomic and
, then it is isomorphic to the space
with
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 -ring)
of subsets of a set
such that
implies
, is called a Boolean algebra or an algebra (respectively, a
-algebra or a
-field, cf. also Algebra of sets). Usually, in a measure space
the
-ring
can be proved to be a
-field (this holds, in particular, if
).
The phrase "totally (s-) finite" is seldom used.
Borel has given very nice ideas in order to construct the measure , but Lebesgue was the first to give a satisfactory construction of it, as a byproduct of the construction of
.
A product space is also often written as a (kind of) tensor product: .
A family of measurable spaces with compatible probability measures on each finite product is called a projective system of measure spaces, and the corresponding probability measure on
, if it exists, is called the projective limit; it exists if
is countable (the Ionescu–Tulcea theorem, cf. [5]).
Suppose that is a topological space and
is its Borel
-field; then
is perfect for every finite measure
if
is a Polish space or, more generally, a Luzin space (in which case
is often called a standard measurable space) or, still more generally, a Suslin space (in which case
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 is the space of rational numbers, or, more generally, when
is a Luzin space which is not Polish. See [a1].
In the abstract setting, whenever is a sequence of finite measures on
, where
is a
-field, such that
![]() |
exists for any , then
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 |
Measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure&oldid=29801