# Convergence of Measures and Metrizability

This article addresses narrow and wide convergence of probability measures.

## Contents

## 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 : a function from a partially ordered set such that for each such that . We say that this net converges to if, for every open set , there exists some large such that for all , .

## Weak-star Topologies

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

In the case where is norm separable, the weak-star topology on the unit ball of can, in fact, be metrized.^{[3]} Fix a sequence that is countable and dense in . Define the metric by . This is a sum of pseudometrics, necessarily convergent because each term is less than or equal to , and is nondegenerate because if , then for each , which would imply that the continuous functions agreed on a dense subset of a metric space. The identity map from to is continuous: choose a convergent net in , . Then for each , perform the following truncation process: choose a large so that . Because , for each , there is some large such that for all , . By the net order axioms, there is some large . So for each , . 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 -spaces

If is a compact Hausdorff metric space, is separable, due to the following argument: compact metric spaces are always separable^{[4]}. Pick a countable dense subset , and consider the smallest -algebra generated by the functions . 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 is metrizable.

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

## Narrow Convergence

For every finite signed Radon measure on a locally compact Hausdorff space , there is some element such that . Moreover, letting denote the total variation of the measure, there is a net of functions such that , and . This means that , 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, . Narrow convergence is weak-star convergence in with respect to : a net of measures converges to narrowly if, for every , ^{[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 is separable. Take, again, a countable dense subset of , , and taking the family of functions . Let be the family of functions generated by taking infima of finitely many elements of , and let . This is still countable, and approximates integrals of elements of well weakly-star, so there is a metric on the probabilities by enumerating , and
^{[8]}.

## Wide Convergence

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

## References

- ↑ Kelley,
*General Topology*Ch. 2. Springer, 1975. - ↑
^{2.0}^{2.1}Kadison, Ringrose.*Fundamentals of the Theory of Operator Algebras, Volume I.*Ch. 1.3, 1.6. Academic Press, 1983. - ↑ Rudin,
*Functional Analysis*Ch. 3. 1991. - ↑ Math StackExchange,
*Prove if is a compact metric space, then is separable.*2020. - ↑ [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. - ↑ Math StackExchange,
*Stone-Cech via .*2020. - ↑
^{7.0}^{7.1}Villani.*Optimal transport, old and new.*2006. - ↑ Ambrosio, Gigli, Savaré.
*Gradient Flows.*Ch. 5.1. 2000