Convergence of measures

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: \[ \abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \text{$\{B_i\}\subset\mathcal{B}$ is a countable partition of $B$}\right\}. \] 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. =='"`UNIQ--h-0--QINU`"'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. ==='"`UNIQ--h-1--QINU`"'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''. ==='"`UNIQ--h-2--QINU`"'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 $\mathcal{M}^b$. ==='"`UNIQ--h-3--QINU`"'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 [[#EG|[EG]]]). ==='"`UNIQ--h-4--QINU`"'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|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|[Bi]]], [[#Ma|[Ma]]] and [[#EG|[EG]]]). This is, however, inconsistent with the terminology of Banach spaces, see below. =='"`UNIQ--h-5--QINU`"'Properties== ==='"`UNIQ--h-6--QINU`"'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. ==='"`UNIQ--h-7--QINU`"'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|[De]]]). ==='"`UNIQ--h-8--QINU`"'Compactness of the weak$^*$ topology=== If $\{\mu_k\}$ is a sequence with $\sup_k \|\mu_k\|_v < \infty$ 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|[EG]]]). =='"`UNIQ--h-9--QINU`"'Probability measures== On the space of probability measures one can get further interesting properties. ==='"`UNIQ--h-10--QINU`"'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|[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]]. ==='"`UNIQ--h-11--QINU`"'[[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 [[#Vi|[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 $\varphi$).

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