Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures?

Question

Let $X$ be a locally compact Hausdorff space. Denote $C_c(X)$ and $C_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation theorem, the dual space $C_0(X)^*$ is isomorphic to the space of finite Radon measures $M(X)$. Since $C_c(X)$ is dense in $C_0(X)$, we actually have $$ C_c(X)^* = C_0(X)^* = M(X) $$ In probability textbooks, the vague topology on $M(X)$, depending on the author, is defined as the weak* topology induced by either $C_c(X)$ or $C_0(X)$. However, these two topologies are in general not the same.

I've heard somewhere that the reason why probabilists don't care about this ambiguity is, because $C_c(X)$ and $C_0(X)$ induce the same vague topology on a closed ball in $M(X)$, and probabilists only care about measures of norm $\leq 1$ anyway.

So is the statement true? And how to prove it?

Answer 1

The dual unit ball of a normed space $E$ is weakly compact à la Alaoglu and hence there is no strictly coarser Hausdorff topology. Applying this to $E=C_0(X)$ you get that the topologies $\sigma(M(X),C_0(X))$ and $\sigma(M(X),C_c(X))$ coincide on all bounded sets of $M(X)$.