Difference between revisions of "Convergence of measures"
(MSC|28A33 Category:Classical measure theory (again)) |
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+ | A concept in measure theory, determined by a certain topology in a space of measures that are defined on a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261401.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261402.png" /> of subsets of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261403.png" /> or, more generally, in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261404.png" /> of charges, i.e. countably-additive real or complex functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261405.png" />, defined on sets from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261406.png" />. The following are the most commonly used topologies in the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261407.png" /> consisting of bounded charges, i.e. charges for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261409.png" />. | ||
+ | |||
+ | 1) In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614010.png" /> the norm | ||
+ | |||
+ | <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/c/c026/c026140/c02614011.png" /></td> </tr></table> | ||
+ | |||
+ | called the variation of the charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614012.png" />, is introduced. The convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614014.png" />, to a charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614015.png" /> in this norm is called convergence in variation. | ||
+ | |||
+ | 2) In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614016.png" /> the ordinary weak topology is examined: Convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614018.png" />, in this topology (weak convergence) means that for any continuous linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614022.png" />. This convergence is equivalent to the fact that the sequence of charges is bounded, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614023.png" />, and that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614024.png" /> the sequence of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614026.png" />. Weak convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614028.png" /> implies convergence of the integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614030.png" />, for any bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614032.png" /> that is measurable with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614033.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614034.png" />. | ||
+ | |||
+ | 3) When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614035.png" /> is a topological space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614036.png" /> is its Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614037.png" />-algebra, a topology is examined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614038.png" /> which is also called the weak topology (or sometimes the narrow topology). It is defined as the weakest of the topologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614039.png" /> relative to which all functionals 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/c/c026/c026140/c02614040.png" /></td> </tr></table> | ||
+ | |||
+ | are continuous, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614041.png" /> is an arbitrary bounded continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614042.png" />. This topology is weaker than the previous one, and convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614044.png" />, relative to it (weak or narrow convergence) is equivalent to the convergence of the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614046.png" />, for any Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614047.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614049.png" /> and the operation of closure of a set is denoted by the bar. | ||
+ | |||
+ | 4) When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614050.png" /> is a locally compact topological space (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614051.png" /> is a Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614052.png" />-algebra) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614053.png" /> the so-called wide topology is examined: the convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614055.png" /> (wide convergence), means convergence of the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614057.png" />, for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614058.png" /> with compact support. This topology is weaker than the weak topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614059.png" />. An analogous topology is defined naturally in the wider space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614060.png" /> of locally bounded charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614061.png" />, i.e. charges such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614062.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614063.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614066.png" />. | ||
+ | |||
+ | ====References==== | ||
+ | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Billingsley, "Convergence of probability measures" , Wiley (1968)</TD></TR></table> |
Revision as of 17:53, 7 February 2012
2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]
A concept in measure theory, determined by a certain topology in a space of measures that are defined on a certain -algebra
of subsets of a space
or, more generally, in a space
of charges, i.e. countably-additive real or complex functions
, defined on sets from
. The following are the most commonly used topologies in the subspace
consisting of bounded charges, i.e. charges for which
,
.
1) In the norm
![]() |
called the variation of the charge , is introduced. The convergence of a sequence of charges
,
, to a charge
in this norm is called convergence in variation.
2) In the ordinary weak topology is examined: Convergence of a sequence of charges
,
, in this topology (weak convergence) means that for any continuous linear function
on
,
,
. This convergence is equivalent to the fact that the sequence of charges is bounded,
, and that for any set
the sequence of values
,
. Weak convergence of a sequence of charges
,
implies convergence of the integrals
,
, for any bounded function
on
that is measurable with respect to the
-algebra
.
3) When is a topological space and
is its Borel
-algebra, a topology is examined in
which is also called the weak topology (or sometimes the narrow topology). It is defined as the weakest of the topologies in
relative to which all functionals of the form
![]() |
are continuous, where is an arbitrary bounded continuous function on
. This topology is weaker than the previous one, and convergence of a sequence of charges
,
, relative to it (weak or narrow convergence) is equivalent to the convergence of the values
,
, for any Borel set
for which
, where
and the operation of closure of a set is denoted by the bar.
4) When is a locally compact topological space (and
is a Borel
-algebra) in
the so-called wide topology is examined: the convergence of a sequence of charges
,
(wide convergence), means convergence of the functionals
,
, for any continuous function
with compact support. This topology is weaker than the weak topology in
. An analogous topology is defined naturally in the wider space
of locally bounded charges
, i.e. charges such that for any point
there is a neighbourhood
in which
,
,
.
References
[1] | N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) |
[2] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |
[3] | P. Billingsley, "Convergence of probability measures" , Wiley (1968) |
Convergence of measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convergence_of_measures&oldid=20872