Convergence of Measures and Metrizability

This article addresses narrow and wide convergence of probability measures.

Weak-star Topologies

Given a Banach space $X$ and its Banach dual $X^{*}$, the dual can be endowed with the weakest topology that makes the evaluation maps at elements of $X$ continuous. This is called the weak-star topology relative to $X$. By Banach-Alaoglu, the unit ball of $X^{*}$ (which we call $(X^{*})_{1}$) with the weak-star topology is compact.

In the case where $X$ is norm separable, the weak-star topology on the unit ball of $X^{*}$ can, in fact, be metrized. Fix a sequence $\{x_{n}\}_{n=1}^{\infty }$ that is countable and dense in $X$. Define the metric $d$ by $d(\phi ,\psi ):=\sum _{n=0}^{\infty }2^{-n}{\frac {|\phi (x_{n})-\psi (x_{n})|}{1+|\phi (x_{n})-\psi (x_{n})|}}$. This is a sum of pseudometrics, necessarily convergent because each term is less than or equal to $2^{-n}$, and is nondegenerate because if $d(\phi ,\psi )=0$, then $\phi (x_{n})=\psi (x_{n})$ for each $x_{n}$, which would imply that the continuous functions $\phi ,\psi$ agreed on a dense subset of a metric space. The identity map from $((X^{*})_{1},w*)$ to $((X^{*})_{1},d)$ is continuous: choose a net $(\phi _{\gamma })_{\gamma \in \Gamma }\to \phi$. Then for each $\epsilon >0$, perform the following truncation process: choose a large $N$ so that $\sum _{n=N+1}^{\infty }2^{-n}=2^{-N}<{\frac {\epsilon }{2}}$. Because $\phi _{\gamma }\xrightarrow {w*} \phi$, for each $n\in \{1,\ldots ,N\}$, there is some large $\gamma _{n}$ such that for all $\gamma \succeq \gamma _{n}$, $|\phi _{\gamma }(x_{n})-\phi (x_{n})|<{\frac {\epsilon }{2\cdot N\cdot 2^{-n}}}$. By the net order axioms, there is some large $\gamma _{0}\succeq \gamma _{i}\forall i\in \{1,\ldots ,N\}$. So for each $\gamma \succeq \gamma _{0}$, $d(\phi _{\gamma },\phi )<\sum _{n=1}^{N}{\frac {1}{2N}}+\sum _{n=N+1}^{\infty }2^{-n}<\epsilon$. Now the identity map between the two spaces is a continuous bijection between a compact and a Hausdorff topological space, and is therefore a homeomorphism. So it metrizes the weak-star topology.

Metrizability for duals of $C(X)$-spaces

If $X$ is a compact Hausdorff metric space, $C(X)=C_{b}(X)=C_{0}(X)$ is separable, due to the following argument: compact metric spaces are always separable. Pick a countable dense subset $\{x_{n}\}_{n\in \mathbb {N} }\subseteq X$, and consider the smallest $\mathbb {Q}$-algebra generated by the functions $d(x,x_{n})$. This is a countable union of countable sets and therefore countable. As a subalgebra which separates points and contains the constant function, it must be dense by Stone-Weierstrass. So the unit ball of the dual of $C(X)$ is metrizable.

Conversely, assume that the unit ball of the dual of $C_{b}(X)$ is metrizable. Then, because $X\rightarrow (C_{b}(X))^{*}$ via point evaluation, which is a norm $1$ map, and because the topology on $X$ is exactly the topology of weak-star convergence in $C_{b}(X)^{*}$, this means that $X$ is metrizable as well. So $X$ is a metrizable space which lives inside a compact metrizable space. This is in fact equivalent to the Stone-Cech compactification of $X$ being metrizable, which is quite rare.

Narrow Convergence

For every finite signed Radon measure $\mu$ on a locally compact Hausdorff space $X$, there is some element $f\in C_{0}(X)$ such that $\int _{X}f\,d\mu \neq 0$. Moreover, letting $|\mu |$ denote the total variation of the measure, there is a net of functions $f_{\gamma }\in C_{0}(X)$ such that $\int _{X}f_{\gamma }\,d\mu \rightarrow |\mu |$, and $\int _{X}f\,d\mu \leq \|f\|_{\infty }|\mu |$. This means that ${\mathcal {M}}(X)\hookrightarrow C_{0}(X)^{*}$, and can be isometrically identified with a subset of the dual. In particular, because the total variation norm does not increase with respect to continuous bounded functions, ${\mathcal {M}}(X)\hookrightarrow C_{b}(X)^{*}$. Narrow convergence is weak-star convergence in ${\mathcal {M}}(X)$ with respect to $C_{b}(X)$: a net of measures $\mu _{\gamma }$ converges to $\mu$ narrowly if, for every $f\in C_{b}(X)$, $\int _{X}f\,d\mu _{n}\to \int _{X}f\,d\mu$.

Although in general the topology of narrow convergence is not metrizable on the unit ball -- for example, on any compact Hausdorff space which is not metrizable -- it is on the class of probability measures, so long as $X$ is separable. Take, again, a countable dense subset of $X$, $D$, and taking the family of functions ${\mathcal {C}}_{2}=\{h(x)=(q_{1}+q_{2}d(x,y))\wedge k\,|\,q_{1},q_{2},k\in \mathbb {Q} ,q_{2},k\in (0,1),y\in D\}$. Let ${\mathcal {C}}_{1}$ be the family of functions generated by taking infima of finitely many elements of ${\mathcal {C}}_{2}$, and let ${\mathcal {C}}_{0}=\{\lambda h\,|\,\lambda \in \mathbb {Q} ,h\in {\mathcal {C}}_{1}$. This is still countable, and approximates integrals of elements of $C_{b}$ well weakly-star, so there is a metric on the probabilities by enumerating ${\mathcal {C}}_{0}=\{f_{k}\}_{k=1}^{\infty }$, and $d(\mu ,\nu )=\sum _{k=1}^{\infty }2^{-k}|\int f_{k}\,d\mu -\int f_{k}\,d\nu |.$

Wide Convergence

Wide convergence is the weak-star convergence with respect to elements of $C_{0}(X)$ rather than $C_{b}(X)$. As such, it is a weaker topology on the class of probability measures. When $X$ is a locally compact separable metric space (in particular, locally compact metric spaces which are $\sigma$-compact are separable), we can find a separating family of functions which form a countable subalgebra. One way to do this is by taking, for each point $x_{n}$ in a countable dense subset, a compact neighborhood $K_{n}\ni x_{n}$ and taking a partition of unity subordinate to that compact, $\varphi _{n}$. Taking the $\mathbb {Q}$-algebra generated by this countable family of functions will separate points, which will make it dense in $C_{0}(X)$ by Stone-Weierstrass. So $C_{0}(X)$ is separable, which means that the unit ball of the dual is metrizable.