# The name and properties of the convergence of measures in the weak topology

Let $X$ be a topological space endowed with the Borel $\sigma$-algebra, and let $\mathcal M(X) \subseteq C_b(X)^*$ be the space of the complex measures on it, with $C_b(X)$ being the bounded continuous functions. Endow $\mathcal M(X)$ with the trace of the weak topology, i.e. $\mu_i \to \mu$ if and only if $T(\mu_i) \to T(\mu) \ \forall T \in C_b(X)^{**}$ (the bidual being a very ugly space, difficult to describe in general).

Does the convergence of measures in this topology have a name? What is the relationship with other forms of convergence of measures?

• Even if $X$ is a relatively nice space (like $[0,1]$ with Borel sets), I don't think it is possible to give an explicit description of the dual of $\mathcal M(X)$, see this question: math.stackexchange.com/questions/47544/… – Shalop Jan 23 '18 at 21:56
• @Shalop: I agree with you, but do I need such a description? It would be convenient to have it, but not necessary in order to consider the convergence in the weak topology as described above. – Alex M. Jan 24 '18 at 9:51
• @AlexM. I would guess Billingsley's text on convergence of measures might have an answer (I've only looked at some parts of it when I've needed a reference.) The most useful topology on $M(X)$ is the weak* topology (as I'm sure you know) and I've heard good probabilists claim that "weak convergence is never studied." – 3-in-441 Jan 25 '18 at 4:57
• @AlexM. How do you classify weak convergence without specifying the dual? Like 3in441 says, one ordinarily uses weak* convergence, i.e. Convergence by elements of the pre-dual which is $C(X)$ when $X$ is nice. – Shalop Jan 27 '18 at 18:09
• @Shalop: I might have been sloppy in my formulation, so I have reformulated my question. – Alex M. Jan 27 '18 at 18:25