A characterization of almost everywhere continuous functions (Q1912737)

Let \((X,d)\) be a separable metric space and \({\mathcal M}(X)\) the set of probability measures on the \(\sigma\)-algebra of Borel sets in \(X\). In this paper, the author proves that a bounded measurable function \(f:X\to\mathbb{R}\) is almost everywhere continuous with respect to \(\mu\in{\mathcal M}(X)\) if and only if \(\lim_{n\to\infty} \int_Xf d\mu_n=\int_Xf d\mu\) for any sequence \(\{\mu_n\}\) in \({\mathcal M}(X)\) weakly convergent to \(\mu\).