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

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?

• Most probabilists will probably want to use all bounded continuous functions and not restrict themselves to locally compact spaces. Feb 18 at 12:15

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 $$M(X)$$.