View source for Convergence of measures

{{MSC|28A33}}

[[Category:Classical measure theory]]

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$\newcommand{\abs}[1]{\left|#1\right|}$

A concept in measure theory, determined by a certain topology in a space of measures that are defined on a certain [[Algebra of sets|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 measure|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.


==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 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 {{Cite|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|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 {{Cite|Bi}}, {{Cite|Ma}} and {{Cite|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|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.

===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 {{Cite|De}}).

===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 {{Cite|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 {{Cite|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 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 {{Cite|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 $\varphi$).

==References==
{|
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|valign="top"|{{Ref|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. {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|valign="top"|{{Ref|Bi}}|| P. Billingsley, "Convergence of  probability measures" , Wiley (1968) {{MR|0233396}} {{ZBL|0172.21201}}
|-
|valign="top"|{{Ref|Bo}}|| N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}}
|-
|valign="top"|{{Ref|De}}|| C. De Lellis, "Rectifiable  sets, densities  and tangent measures" Zurich Lectures in Advanced  Mathematics. European  Mathematical Society (EMS), Zürich, 2008.  {{MR|2388959}}  {{ZBL|1183.28006}}
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|valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}}
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|valign="top"|{{Ref|EG}}|| L.C. Evans,  R.F. Gariepy, "Measure theory  and fine properties of functions"  Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,  1992.  {{MR|1158660}} {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Ma}}|| P. Mattila, "Geometry of sets and measures in euclidean spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
|-
|valign="top"|{{Ref|Vi}}|| Villani, Cédric Topics in optimal transportation.Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI,  2003. {{MR|1964483}} {{ZBL|1106.90001}} 
|}