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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" />.
+
[[Category:Classical measure theory]]
  
1) In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614010.png" /> the norm
+
{{TEX|done}}
 +
$\newcommand{\abs}[1]{\left|#1\right|}$
  
<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>
+
A concept in measure theory, determined by a certain topology in a space of measures that are defined on a certain [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of subsets of a space $X$ or, more generally, in a space $\mathcal{M} (X, \mathcal{B})$ of charges, i.e. countably-additive real (resp. complex) functions $\mu: \mathcal{B}\to \mathbb R$ (resp. $\mathbb C$), often also called $\mathbb R$ (resp. $\mathbb C$) valued or [[Signed measure|signed measures]]. The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as:
 +
\[
 +
\abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \{B_i\}\subset\mathcal{B} \text{ is a countable partition of } B \right\}.
 +
\]
 +
In the real-valued case the above definition simplifies as
 +
\[
 +
\abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (B\setminus A)}\right).
 +
\]
 +
The total variation of $\mu$ is then defined as $\left\|\mu\right\|_v :=
 +
\abs{\mu}(X)$.
  
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.
+
''Warning'': If $\mathcal{B}$ is the $\sigma$-algebra of Borel sets of a topological space $X$, we will then denote by $\mathcal{M}^b (X)$ the space of ''Radon'' signed measures, i.e. those signed measures with finite total variation such that $|\mu|$ is a [[Radon measure]]. This is actually not a restriction in many cases, for instance if $X$ is the euclidean space.
  
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
+
==Notions of convergence==
 +
The space $\mathcal{M}^b (X, \mathcal{B})$ of $\mathbb R$ (resp. $\mathbb C$) valued measure with finite total variation is a [[Banach space]] and the following are the most commonly used topologies.
  
<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>
+
===The norm or [[strong topology]]===
 +
$\mu_n\to \mu$ if and only if $\left\|\mu_n-\mu\right\|_v\to 0$. This convergence is sometimes called ''convergence in variation''.
  
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.
+
===The [[weak topology]]===  
 +
A sequence of measures $\mu_n \rightharpoonup \mu$ if and only if $F (\mu_n)\to F(\mu)$
 +
for every bounded linear functional $F$ on $\mathcal{M}^b$.
  
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" />.
+
===The narrow topology===
 +
When $X$ is a [[topological space]] and $\mathcal{B}$ the corresponding $\sigma$-algebra of [[Borel set|Borel sets]], we can introduce on $X$ the narrow topology. In this case $\mu_n$ converges to $\mu$ if and only if
 +
\begin{equation}\label{e:narrow}
 +
\int f\, \mathrm{d}\mu_n \to \int f\, \mathrm{d}\mu
 +
\end{equation}
 +
for every bounded continuous function $f:X\to \mathbb R$ (resp. $\mathbb C$). The following is an important consequence of the narrow convergence when $X$ is a [[Locally compact space|locally compact]] [[Hausdorff space]]: if $\mu_n$ converges narrowly to $\mu$, then $\mu_n (A)\to \mu (A)$ for any Borel set such that $\abs{\mu}(\partial A) = 0$ (cp. with Theorem 1(iii) of Section 1.9 in {{Cite|EG}}).
  
====References====
+
===The wide or weak$^\star$ topology===
<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>
+
When $X$ is a locally compact topological space and $\mathcal{B}$ the $\sigma$-algebra of Borel sets yet another topology can be introduced, the so-called wide topology, or sometimes referred to as [[weak-star topology|weak$^\star$ topology]]. A sequence $\mu_n\rightharpoonup^\star \mu$ if and only if \eqref{e:narrow} holds for continuous functions which are compactly supported.
 +
This topology is in general weaker than the narrow topology but they do coincide when restricted to probability measures if $X$ is a Hausdorff space.
 +
 
 +
A topology analogous to the weak$^\star$ topology is defined in the more  general space $\mathcal{M}^b_{loc}$ of locally bounded measures, i.e.  those measures $\mu$ such that for any point $x\in X$ there is a  neighborhood $U$ with $\abs{\mu}(U)<\infty$.
 +
 
 +
'''Warning''' Sequences of measures converging in the narrow (or in the wide topology) are called ''weakly convergent'' sequences by several authors (cp. with {{Cite|Bi}}, {{Cite|Ma}} and {{Cite|EG}}). This is, however, inconsistent with the terminology of Banach spaces, see below.
 +
 
 +
==Properties==
 +
===Relation with functional analysis===
 +
If $X$ is compact and Hausdorff the [[Riesz representation theorem]] shows that $\mathcal{M}^b (X)$ is the dual of the space $C(X)$ of continuous functions. Under this assumption the narrow and weak$^\star$ topology coincides with the usual [[weak-star topology|weak$^\star$ topology]] of the Banach space theory. Since in general $C(X)$ is not a [[Reflexive space|reflexive space]], it turns out that the narrow topology is in general weaker than the weak topology.
 +
 
 +
===Metrizability of the weak$^*$ topology===
 +
On bounded subsets of $\mathcal{M}^b (X)$, the weak$^*$ topology is metrizable. If $X$ is compact, this follows directly from standard functional-analytic arguments, since $\mathcal{M}^b (X)$ is then the dual of a separable Banach space. The case of a $\sigma$-compact $X$ can be reduced to that of a compact space by exhaustion with compact sets.
 +
 
 +
The cone of nonnegative measures is metrizable without further restrictions on the size of the measures (see for instance Proposition 2.6 of {{Cite|De}}).
 +
 
 +
===Compactness of the weak$^*$ topology===
 +
If $\{\mu_k\}$ is a sequence with $\sup_k \|\mu_k\|_v < \infty$ and $X$ is $\sigma$-compact then a subsequence converges weakly$^*$. This is again a consequence of standard Banach space theory if $X$ is compact (see [[Banach-Alaoglu theorem]]), whereas the locally compact case can easily reduced to the compact one by exhaustion. More general compactness statements are possible (cp. for instance with Theorem 2 in Section 1.9 of {{Cite|EG}}).
 +
 
 +
==Probability measures==
 +
On the space of probability measures one can get further interesting properties.
 +
 
 +
===Narrow and wide topology===
 +
The narrow and wide topology coincide on the space of probability measures on a locally compact spaces. If $X$ is compact, then the space of probability measures with the narrow (or wide) topology is also compact. However, if $X$ is not compact, the compactness of the wide topology fails: as an example take the sequence of Dirac masses $\delta_n$ on $\mathbb R$, where $n\in \mathbb N$. This sequence converges, in the wide topology, to the measure $0$. However, if one assumes ''tightness'' of the sequence of measures $\{\mu_n\}$ (cp. with \ref{e:tight}), then the sequential (pre)compactness is reestablished. More precisely (cp. with Theorem 6.1 of {{Cite|Bi}}):
 +
 
 +
'''Theorem (Prohorov)'''
 +
Let $X$ be a locally compact Hausdorff space and $\{\mu_k\}$ a sequence of Radon probability measures. If
 +
\begin{equation}\label{e:tight}
 +
\forall \varepsilon\; \exists K\, \mbox{compact such that }\; \mu_k (X\setminus K)<\varepsilon \; \forall k\,
 +
\end{equation}
 +
then a subsequence converges weakly$^*$ to a probability Radon measure $\mu$.
 +
 
 +
A sequence of probability measures converging in the narrow topology is often called a ''weakly converging sequence''.
 +
See [[Weak convergence of probability measures]].
 +
 
 +
===[[Wasserstein metric|Wasserstein metrics]]===
 +
The space of probability measures on a [[Polish space]] can be endowed with several interesting metrics, called [[Wasserstein metric|Wasserstein]] or Monge-Kantorovich distances (see Section 7.1 of {{Cite|Vi}}) and related to the [[Mass transport]] problem. The $1$-Wasserstein distance (also called Kantorovich-Rubinstein distance) is defined as
 +
\[
 +
W_1 (\mu, \nu) = \sup \left\{ \int \varphi d\mu - \int \varphi d\nu : \; \varphi\in C(X, \mathbb R)\; \mbox{ with }{\rm Lip}\, (\varphi)\leq 1 \right\}
 +
\]
 +
(here ${\rm Lip}\, (\varphi)$ denotes the [[Lipschitz condition|Lipschitz constant]] of $\varphi$).
 +
 
 +
==References==
 +
{|
 +
|-
 +
|valign="top"|{{Ref|AmFuPa}}|| L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. {{MR|1857292}}{{ZBL|0957.49001}}
 +
|-
 +
|valign="top"|{{Ref|Bi}}|| P. Billingsley, "Convergence of probability measures" , Wiley (1968) {{MR|0233396}} {{ZBL|0172.21201}}
 +
|-
 +
|valign="top"|{{Ref|Bo}}|| 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}}
 +
|-
 +
|valign="top"|{{Ref|De}}|| C. De Lellis, "Rectifiable  sets, densities  and tangent measures" Zurich Lectures in Advanced  Mathematics. European  Mathematical Society (EMS), Zürich, 2008.  {{MR|2388959}}  {{ZBL|1183.28006}}
 +
|-
 +
|valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}}
 +
|-
 +
|valign="top"|{{Ref|EG}}|| L.C. Evans,  R.F. Gariepy, "Measure theory  and fine properties of functions"  Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,  1992.  {{MR|1158660}} {{ZBL|0804.2800}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}|| P. Mattila, "Geometry of sets and measures in euclidean spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
 +
|-
 +
|valign="top"|{{Ref|Vi}}|| Villani, Cédric Topics in optimal transportation.Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI2003. {{MR|1964483}} {{ZBL|1106.90001}}
 +
|}

Latest revision as of 08:33, 16 August 2013

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

A concept in measure theory, determined by a certain topology in a space of measures that are defined on a certain σ-algebra $\mathcal{B}$ of subsets of a space $X$ or, more generally, in a space $\mathcal{M} (X, \mathcal{B})$ of charges, i.e. countably-additive real (resp. complex) functions $\mu: \mathcal{B}\to \mathbb R$ (resp. $\mathbb C$), often also called $\mathbb R$ (resp. $\mathbb C$) valued or signed measures. The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \{B_i\}\subset\mathcal{B} \text{ is a countable partition of } B \right\}. \] In the real-valued case the above definition simplifies as \[ \abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (B\setminus A)}\right). \] The total variation of $\mu$ is then defined as $\left\|\mu\right\|_v := \abs{\mu}(X)$.

Warning: If $\mathcal{B}$ is the $\sigma$-algebra of Borel sets of a topological space $X$, we will then denote by $\mathcal{M}^b (X)$ the space of Radon signed measures, i.e. those signed measures with finite total variation such that $|\mu|$ is a Radon measure. This is actually not a restriction in many cases, for instance if $X$ is the euclidean space.


Notions of convergence

The space $\mathcal{M}^b (X, \mathcal{B})$ of $\mathbb R$ (resp. $\mathbb C$) valued measure with finite total variation is a Banach space and the following are the most commonly used topologies.

The norm or strong topology

$\mu_n\to \mu$ if and only if $\left\|\mu_n-\mu\right\|_v\to 0$. This convergence is sometimes called convergence in variation.

The weak topology

A sequence of measures $\mu_n \rightharpoonup \mu$ if and only if $F (\mu_n)\to F(\mu)$ for every bounded linear functional $F$ on $\mathcal{M}^b$.

The narrow topology

When $X$ is a topological space and $\mathcal{B}$ the corresponding $\sigma$-algebra of Borel sets, we can introduce on $X$ the narrow topology. In this case $\mu_n$ converges to $\mu$ if and only if \begin{equation}\label{e:narrow} \int f\, \mathrm{d}\mu_n \to \int f\, \mathrm{d}\mu \end{equation} for every bounded continuous function $f:X\to \mathbb R$ (resp. $\mathbb C$). The following is an important consequence of the narrow convergence when $X$ is a locally compact Hausdorff space: if $\mu_n$ converges narrowly to $\mu$, then $\mu_n (A)\to \mu (A)$ for any Borel set such that $\abs{\mu}(\partial A) = 0$ (cp. with Theorem 1(iii) of Section 1.9 in [EG]).

The wide or weak$^\star$ topology

When $X$ is a locally compact topological space and $\mathcal{B}$ the $\sigma$-algebra of Borel sets yet another topology can be introduced, the so-called wide topology, or sometimes referred to as weak$^\star$ topology. A sequence $\mu_n\rightharpoonup^\star \mu$ if and only if \eqref{e:narrow} holds for continuous functions which are compactly supported. This topology is in general weaker than the narrow topology but they do coincide when restricted to probability measures if $X$ is a Hausdorff space.

A topology analogous to the weak$^\star$ topology is defined in the more general space $\mathcal{M}^b_{loc}$ of locally bounded measures, i.e. those measures $\mu$ such that for any point $x\in X$ there is a neighborhood $U$ with $\abs{\mu}(U)<\infty$.

Warning Sequences of measures converging in the narrow (or in the wide topology) are called weakly convergent sequences by several authors (cp. with [Bi], [Ma] and [EG]). This is, however, inconsistent with the terminology of Banach spaces, see below.

Properties

Relation with functional analysis

If $X$ is compact and Hausdorff the Riesz representation theorem shows that $\mathcal{M}^b (X)$ is the dual of the space $C(X)$ of continuous functions. Under this assumption the narrow and weak$^\star$ topology coincides with the usual weak$^\star$ topology of the Banach space theory. Since in general $C(X)$ is not a reflexive space, it turns out that the narrow topology is in general weaker than the weak topology.

Metrizability of the weak$^*$ topology

On bounded subsets of $\mathcal{M}^b (X)$, the weak$^*$ topology is metrizable. If $X$ is compact, this follows directly from standard functional-analytic arguments, since $\mathcal{M}^b (X)$ is then the dual of a separable Banach space. The case of a $\sigma$-compact $X$ can be reduced to that of a compact space by exhaustion with compact sets.

The cone of nonnegative measures is metrizable without further restrictions on the size of the measures (see for instance Proposition 2.6 of [De]).

Compactness of the weak$^*$ topology

If $\{\mu_k\}$ is a sequence with $\sup_k \|\mu_k\|_v < \infty$ and $X$ is $\sigma$-compact then a subsequence converges weakly$^*$. This is again a consequence of standard Banach space theory if $X$ is compact (see Banach-Alaoglu theorem), whereas the locally compact case can easily reduced to the compact one by exhaustion. More general compactness statements are possible (cp. for instance with Theorem 2 in Section 1.9 of [EG]).

Probability measures

On the space of probability measures one can get further interesting properties.

Narrow and wide topology

The narrow and wide topology coincide on the space of probability measures on a locally compact spaces. If $X$ is compact, then the space of probability measures with the narrow (or wide) topology is also compact. However, if $X$ is not compact, the compactness of the wide topology fails: as an example take the sequence of Dirac masses $\delta_n$ on $\mathbb R$, where $n\in \mathbb N$. This sequence converges, in the wide topology, to the measure $0$. However, if one assumes tightness of the sequence of measures $\{\mu_n\}$ (cp. with \ref{e:tight}), then the sequential (pre)compactness is reestablished. More precisely (cp. with Theorem 6.1 of [Bi]):

Theorem (Prohorov) Let $X$ be a locally compact Hausdorff space and $\{\mu_k\}$ a sequence of Radon probability measures. If \begin{equation}\label{e:tight} \forall \varepsilon\; \exists K\, \mbox{compact such that }\; \mu_k (X\setminus K)<\varepsilon \; \forall k\, \end{equation} then a subsequence converges weakly$^*$ to a probability Radon measure $\mu$.

A sequence of probability measures converging in the narrow topology is often called a weakly converging sequence. See Weak convergence of probability measures.

Wasserstein metrics

The space of probability measures on a Polish space can be endowed with several interesting metrics, called Wasserstein or Monge-Kantorovich distances (see Section 7.1 of [Vi]) and related to the Mass transport problem. The $1$-Wasserstein distance (also called Kantorovich-Rubinstein distance) is defined as \[ W_1 (\mu, \nu) = \sup \left\{ \int \varphi d\mu - \int \varphi d\nu : \; \varphi\in C(X, \mathbb R)\; \mbox{ with }{\rm Lip}\, (\varphi)\leq 1 \right\} \] (here ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$).

References

[AmFuPa] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[Bo] 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
[De] C. De Lellis, "Rectifiable sets, densities and tangent measures" Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. MR2388959 Zbl 1183.28006
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Vi] Villani, Cédric Topics in optimal transportation.Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. MR1964483 Zbl 1106.90001
How to Cite This Entry:
Convergence of measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convergence_of_measures&oldid=20873
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article