Interior points of convex compactа and continuous choice of exact measures

For a metric space $M$ we prove existence of continuous maps $\{M_n\}^{\infty}_{n=1}$ associating to a compact subset $K \subset M$ a probability measure $M_n(K)$ with $\operatorname{supp}(M_n(K)) = K$ in such a way that the set $\{M_n(K)\}^{\infty}_{n=1}$ is dense in the space of probability measures on $K$.