Convergence of Measures and Metrizability

This article addresses narrow and wide convergence of probability measures.

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. Roughly, a net consists of a directed set \(\{x_\gamma\}_{\gamma\in\Gamma}\): a function from a partially ordered set \(\Gamma\) such that for each \(\gamma_1,\gamma_2\in\Gamma,\exists \gamma_3\in\Gamma\) such that \(\gamma_1,\gamma_2\preceq\gamma_3\). We say that this net converges to \(x\) if, for every open set \(U\ni x\), there exists some large \(a_U\) such that for all \(a_U\preceq a\), \(x_{\gamma_a}\in U\).

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-* topology on \(X^*\) 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 convergent net in \((X^*)_1\), \((\phi_\gamma)_{\gamma\in\Gamma} \to \phi\). 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, let \(X\) be a locally compact Hausdorff topological space, and 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 \(\int_X f\,d\mu\le \|f\|_\infty |\mu|\). 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 \(\mathcal C_2=\{h(x)=(q_1+q_2d(x,y))\wedge k\,|\, q_1,q_2,k\in\mathbb Q, q_2,k\in (0,1), y\in D\}\). 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.

References

Kadison, Ringrose. Fundamentals of the Theory of Operator Algebras, Volume I. Ch. 1.3, 1.6. Academic Press, 1983. Kelley, General Topology Ch. 2. Springer, 1975. Rudin, Functional Analysis Ch. 3. 1991. Math StackExchange, Prove if \(X\) is a compact metric space, then \(X\) is separable. 2020. Math StackExchange, Stone-Cech via \(C_b(X)\cong C(\beta X)\). 2020. Ambrosio, Gigli, Savaré. Gradient Flows. Ch. 5.1. 2000 [http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=6155&option_lang=eng Gelfand, Naimark. On the imbedding of normed rings into the ring of operators on Hilbert space. 1943. Villani. Optimal transport, old and new. 2006.