Uniform Convergence of Metrics on Alexandrov Surfaces with Bounded Integral Curvature

We prove uniform convergence of metrics $g_k$ on a closed surface with bounded integral curvature (measure) in the sense of A.D. Alexandrov, under the assumption that the curvature measures $\K_{g_k}=\mu^1_k-\mu^2_k$, where $\mu^1_k,\mu^2_k$ are nonnegative Radon measures that converge weakly to measures $\mu^1,\mu^2$ respectively, and $\mu^1$ is less than $2\pi$ at each point. This generalizes Yu. G. Reshetnyak's well-known result on uniform convergence of metrics on a domain in $\C$, and answers affirmatively the open question on the metric convergence on a closed surface.