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 $\sigma$-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 $\mu: \mathcal{B}\to \mathbb R$ (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 \[ \abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (X\setminus B)}\right). \] The total variation of $\mu$ is then defined as $\left\|\mu\right\|_v := \abs{\mu}(X)$. 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 $\left\|\mu_n-\mu\right\|_v\to 0$. 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 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. $\mathbb C$). This topology is also sometimes called the 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 $\abs{\mu}(\partial A) = 0$. 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|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 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, 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 $\abs{\mu}(U)<\infty$.

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
[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
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Bi]	P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[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