Let $(X,\mathcal{B})$ be a Hausdorff topological space with its Borel $\sigma$-algebra.
What are some general conditions we could impose on $X$ so that finitely supported measures (i.e. finite affine combinations of dirac measures) are weakly dense in the space $\mathcal{M}(X)$ of Borel probability measures of $X$?
Also, it is not hard to prove that if $X$ is a Polish space, then for each Borel probability measure $\mu$ in $X$ there is a sequence $(\mu_n)_n$ of finitely supported measures weakly convergent to $\mu$, but could someone provide a reference so that I can just quote it?
Context: At some point in my research I need to weakly approximate measures in the Pontryagin dual of a countable discrete group by sequences finitely supported measures. So the result for polish spaces is enough for me, but I was wondering how generally can we approximate measures by finitely supported ones.
I don't think there are any real conditions you have to impose. It holds if $X$ is just a topological space and $M(X)$ is the set of Baire measures on $X$ equipped with its weak topology. This is discussed in Chapter 8 of Bogachev Volume II (https://link.springer.com/book/10.1007/978-3-540-34514-5). The relevant things you need to look at are : Definition 8.1.2 and Example 8.1.6 (i).
The only thing you need to enforce is that the Baire sigma algebra is the same as the Borel sigma algebra (and so Baire and Borel measures are the same). This is discussed in Chapter 6, Section 6.3 of the same book. The relevant results are Proposition 6.3.4 and Corollary 6.3.5. It is true for example if $X$ is metric or, more generally, if $X$ is perfectly normal. So Polish is definitely fine.