# Convergence of Measures and Metrizability

## Nets

When speaking about general topological spaces that are not metric spaces, understanding the convergence of sequences does not determine topology or continuity. A generalization of sequences, known as nets, can be used to show continuity.[1] Roughly, a net consists of a directed set ${\displaystyle \{x_{\gamma }\}_{\gamma \in \Gamma }}$: a function from a partially ordered set ${\displaystyle \Gamma }$ such that for each ${\displaystyle \gamma _{1},\gamma _{2}\in \Gamma ,\exists \gamma _{3}\in \Gamma }$ such that ${\displaystyle \gamma _{1},\gamma _{2}\preceq \gamma _{3}}$. We say that this net converges to ${\displaystyle x}$ if, for every open set ${\displaystyle U\ni x}$, there exists some large ${\displaystyle a_{U}}$ such that for all ${\displaystyle a_{U}\preceq a}$, ${\displaystyle x_{\gamma _{a}}\in U}$.

## Weak-star Topologies

Given a Banach space ${\displaystyle X}$ and its Banach dual ${\displaystyle X^{*}}$, the dual can be endowed with the weakest topology that makes the evaluation maps at elements of ${\displaystyle X}$ continuous. This is called the weak-* topology on ${\displaystyle X^{*}}$ relative to ${\displaystyle X}$[2]. By Banach-Alaoglu[2], the unit ball of ${\displaystyle X^{*}}$ (which we call ${\displaystyle (X^{*})_{1}}$) with the weak-star topology is compact.

In the case where ${\displaystyle X}$ is norm separable, the weak-star topology on the unit ball of ${\displaystyle X^{*}}$ can, in fact, be metrized.[3] Fix a sequence ${\displaystyle \{x_{n}\}_{n=1}^{\infty }}$ that is countable and dense in ${\displaystyle X}$. Define the metric ${\displaystyle d}$ by ${\displaystyle d(\phi ,\psi ):=\sum _{n=0}^{\infty }2^{-n}{\frac {|\phi (x_{n})-\psi (x_{n})|}{1+|\phi (x_{n})-\psi (x_{n})|}}}$. This is a sum of pseudometrics, necessarily convergent because each term is less than or equal to ${\displaystyle 2^{-n}}$, and is nondegenerate because if ${\displaystyle d(\phi ,\psi )=0}$, then ${\displaystyle \phi (x_{n})=\psi (x_{n})}$ for each ${\displaystyle x_{n}}$, which would imply that the continuous functions ${\displaystyle \phi ,\psi }$ agreed on a dense subset of a metric space. The identity map from ${\displaystyle ((X^{*})_{1},w*)}$ to ${\displaystyle ((X^{*})_{1},d)}$ is continuous: choose a convergent net in ${\displaystyle (X^{*})_{1}}$, ${\displaystyle (\phi _{\gamma })_{\gamma \in \Gamma }\to \phi }$. Then for each ${\displaystyle \epsilon >0}$, perform the following truncation process: choose a large ${\displaystyle N}$ so that ${\displaystyle \sum _{n=N+1}^{\infty }2^{-n}=2^{-N}<{\frac {\epsilon }{2}}}$. Because ${\displaystyle \phi _{\gamma }\xrightarrow {w*} \phi }$, for each ${\displaystyle n\in \{1,\ldots ,N\}}$, there is some large ${\displaystyle \gamma _{n}}$ such that for all ${\displaystyle \gamma \succeq \gamma _{n}}$, ${\displaystyle |\phi _{\gamma }(x_{n})-\phi (x_{n})|<{\frac {\epsilon }{2\cdot N\cdot 2^{-n}}}}$. By the net order axioms, there is some large ${\displaystyle \gamma _{0}\succeq \gamma _{i}\forall i\in \{1,\ldots ,N\}}$. So for each ${\displaystyle \gamma \succeq \gamma _{0}}$, ${\displaystyle d(\phi _{\gamma },\phi )<\sum _{n=1}^{N}{\frac {1}{2N}}+\sum _{n=N+1}^{\infty }2^{-n}<\epsilon }$. 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 ${\displaystyle C(X)}$-spaces

If ${\displaystyle X}$ is a compact Hausdorff metric space, ${\displaystyle C(X)=C_{b}(X)=C_{0}(X)}$ is separable, due to the following argument: compact metric spaces are always separable[4]. Pick a countable dense subset ${\displaystyle \{x_{n}\}_{n\in \mathbb {N} }\subseteq X}$, and consider the smallest ${\displaystyle \mathbb {Q} }$-algebra generated by the functions ${\displaystyle d(x,x_{n})}$. 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 ${\displaystyle C(X)}$ is metrizable.

Conversely, let ${\displaystyle X}$ be a locally compact Hausdorff topological space, and assume that the unit ball of the dual of ${\displaystyle C_{b}(X)}$ is metrizable. Then, because ${\displaystyle X\rightarrow (C_{b}(X))^{*}}$ via point evaluation, which is a norm ${\displaystyle 1}$ map, and because the topology on ${\displaystyle X}$ is exactly the topology of weak-star convergence in ${\displaystyle C_{b}(X)^{*}}$[5], this means that ${\displaystyle X}$ is metrizable as well. So ${\displaystyle X}$ is a metrizable space which lives inside a compact metrizable space. This is in fact equivalent to the Stone-Cech compactification of ${\displaystyle X}$ being metrizable [6], which is quite rare.

## Narrow Convergence

For every finite signed Radon measure ${\displaystyle \mu }$ on a locally compact Hausdorff space ${\displaystyle X}$, there is some element ${\displaystyle f\in C_{0}(X)}$ such that ${\displaystyle \int _{X}f\,d\mu \neq 0}$. Moreover, letting ${\displaystyle |\mu |}$ denote the total variation of the measure, there is a net of functions ${\displaystyle f_{\gamma }\in C_{0}(X)}$ such that ${\displaystyle \int _{X}f_{\gamma }\,d\mu \rightarrow |\mu |}$, and ${\displaystyle \int _{X}f\,d\mu \leq \|f\|_{\infty }|\mu |}$. This means that ${\displaystyle {\mathcal {M}}(X)\hookrightarrow C_{0}(X)^{*}}$, 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, ${\displaystyle {\mathcal {M}}(X)\hookrightarrow C_{b}(X)^{*}}$. Narrow convergence is weak-star convergence in ${\displaystyle {\mathcal {M}}(X)}$ with respect to ${\displaystyle C_{b}(X)}$: a net of measures ${\displaystyle \mu _{\gamma }}$ converges to ${\displaystyle \mu }$ narrowly if, for every ${\displaystyle f\in C_{b}(X)}$, ${\displaystyle \int _{X}f\,d\mu _{n}\to \int _{X}f\,d\mu }$[7].

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 ${\displaystyle X}$ is separable. Take, again, a countable dense subset of ${\displaystyle X}$, ${\displaystyle D}$, and taking the family of functions ${\displaystyle {\mathcal {C}}_{2}=\{h(x)=(q_{1}+q_{2}d(x,y))\wedge k\,|\,q_{1},q_{2},k\in \mathbb {Q} ,q_{2},k\in (0,1),y\in D\}}$. Let ${\displaystyle {\mathcal {C}}_{1}}$ be the family of functions generated by taking infima of finitely many elements of ${\displaystyle {\mathcal {C}}_{2}}$, and let ${\displaystyle {\mathcal {C}}_{0}=\{\lambda h\,|\,\lambda \in \mathbb {Q} ,h\in {\mathcal {C}}_{1}}$. This is still countable, and approximates integrals of elements of ${\displaystyle C_{b}}$ well weakly-star, so there is a metric on the probabilities by enumerating ${\displaystyle {\mathcal {C}}_{0}=\{f_{k}\}_{k=1}^{\infty }}$, and ${\displaystyle d(\mu ,\nu )=\sum _{k=1}^{\infty }2^{-k}|\int f_{k}\,d\mu -\int f_{k}\,d\nu |}$[8].

## Wide Convergence

Wide convergence is the weak-star convergence with respect to elements of ${\displaystyle C_{0}(X)}$ rather than ${\displaystyle C_{b}(X)}$[7]. As such, it is a weaker topology on the class of probability measures. When ${\displaystyle X}$ is a locally compact separable metric space (in particular, locally compact metric spaces which are ${\displaystyle \sigma }$-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 ${\displaystyle x_{n}}$ in a countable dense subset, a compact neighborhood ${\displaystyle K_{n}\ni x_{n}}$ and taking a partition of unity subordinate to that compact, ${\displaystyle \varphi _{n}}$. Taking the ${\displaystyle \mathbb {Q} }$-algebra generated by this countable family of functions will separate points, which will make it dense in ${\displaystyle C_{0}(X)}$ by Stone-Weierstrass. So ${\displaystyle C_{0}(X)}$ is separable, which means that the unit ball of the dual is metrizable.