I have a sequence of Radon measures (on some set $X$, compact subspace of $\mathbb{R}^d$ so nothing too fancy), say $\mu_n$, which are actually $L^1(X)$ functions. In the limit I want to prove that I obtain a measure supported on a finite set $\{x_1, \ldots, x_N\}$.

One way to prove this seems to be taking a (continuous) function $g$ which is supported outside this set and prove that $\left\langle \mu_n, g \right\rangle$ converges to $0$ as $n$ goes to $+\infty$.

So here are my two questions:

do you have any reference explaining why this is sufficient? I found tons of definitions of the support of a measure, but always in a (much) too general manner.

once I have proved this result, it seems to be also asserted in several publications that the limit must be a sum of Dirac masses located on the set. Do you have any reference for the fact that Radon measures supported on a finite set must be sums of Dirac masses on these points?