Integral Convergence Related to Weak Convergence of Measures

We consider probability measures µn,µon a metric space X such that µn weakly converges to µ. The following convergence lim n→∞ X fn(x)µn(dx )= X f(x)µ(dx) is proved under some restrictions on real valued functions fn and f which are measurable, not necessarily continuous nor bounded. Mathematics Subject Classification: 60F05