Interior Points of Convex Compact and Continuous Selections of Exact Measures

For a metric space \(M\), we prove existence of continuous maps \(\{M_n\}^{\infty}_{n=1}\) associating to each compact set \(K \subset M\), 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 \(K\).