# Uniqueness of the limit of a set of probability measures (discrete)

Let $$K subset mathbb {R} ^ n$$ be compact and consider a sequence of probability measures $$mu_n$$ sure $$K$$. Being weakly bounded, we know that there is a weakly convergent subsequence of the $$mu_n$$. By weak convergence, I mean the convergence given by the action against $$C (K)$$. My question is what kind of conditions can be imposed on the sequence $$mu_n$$ to ensure that the original sequence converges as opposed to a subsequence?

In relation, let me change the question by asking now that the $$mu_n$$ to be discreet that is to say $$mu_n = frac {1} {n} sum_ {i = 1} ^ n delta_ {x_i}$$ for a set (possibly according to n) of points $$x_1, …, x_n in K$$.