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: \[ \abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \{B_i\}\subset\mathcal{B} \text{ is a countable partition of } B \right\}. \] 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

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