Convergence of measures

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 $\sigma$-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 $\mu: \mathcal{B}\mapsto \mathbb R$ (resp. $\mathbb C$), often called also $\mathbb R$(resp. $\mathbb C$)-valued or signed measures. The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as: \[ |\mu| (B) :=\sup\, \left\{ \sum |\mu (B_i)|: \{B_i\}\subset \mathcal{B} \mbox{ is a countable partition of '"`UNIQ-MathJax12-QINU`"'}\right\}\, . \] In the real-valued case the above definition simplies as \[ |\mu| (B) = \sup_{A\in \mathcal{B}, A\subset B}\, \left(|\mu (A)| + |\mu (X\setminus B)|\right)\, . \] The total variation of $\mu$ is then defined as $\|\mu\|_v := |\mu| (X)$. 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.

1) The norm or strong topology: $\mu_n\to \mu$ if and only if $\|\mu_n-\mu\|_v\to 0$.

2) 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 $\mathcal{M}^b$.

3) 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\, d\mu_n \to \int f\, d\mu \end{equation} for every bounded continuous function $f:X\to \mathbb R$(resp. $\mathbb C$). This topology is also called sometimes weak topology, however such notation is inconsistent with the Banach space theory, see below. The following is an important consequence of the narrow convergence: if $\mu_n$ converges narrowly to $\mu$, then $\mu_n (A)\to \mu (A)$ for any Borel set such that $|\mu| (\partial A)=0$.

4) 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. If $X$ is compact and Hausdorff the Riesz representation theorem shows that $\mathcal{M}^b$ is the dual of the space $C(X)$ of continuous functions. Under this assumption the narrow and weak$^\star$ topology coincide 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.

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 $|\mu| (U)<\infty$.

http://www.encyclopediaofmath.org/index.php/Convergence_of_measures

References

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