Consider a probability distribution on $\mathbb{R}^k$, say $\mu$. Then there is a sequence of probability measures $\mu\_n$ that converge weakly to $\mu$ so that each of them is discrete (takes finitely values).


Question: Assume we replace $\mathbb{R}^k$ by any any locally compact Hausdorf group $G$. Is the previous still valid? If so, could anyone please provide a useful reference? 



Discretizing probability measures
Since the answer above addresses the problem for signed measures rather than probabilities let me add to it by noting that both versions are true and are direct consequences of versions of the Hahn-Banach theorem (bipolar theorem). The appropriate duality is that between the space of bounded, continuous functions on a completely regular space and that of the bounded, Radon measures thereon. Thus neither local compactness nor a group structure is directly relevant.


