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?



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


