Convergence of Measures and Metrizability

This article should address metrizability for both narrow and wide convergence.

General Functional Analysis Refs

• Ambrosio, Gigli, Savaré (107-108), Brezis (72-76)

Narrow Convergence

• Unit ball of dual space of $C_{b}(X)$ is, in general, not metrizable: [Conway, A Course in Functional Analysis, Theorem 6.6]
• It is metrizable when restricted to probability measures: Ambrosio, Gigli, Savaré (106-108)

Wide Convergence

• On the other hand, if X is a metrizable locally compact space that is σ-compact, then $C_{0}(X)$ is separable, [Conway, A Course in Functional Analysis, III. Banach Spaces, exercise 14]