# Convergence of measures

2010 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] $\newcommand{\abs}{\left|#1\right|}$

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 $\varphi$).

How to Cite This Entry:
Convergence of measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convergence_of_measures&oldid=30088
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article