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We prove uniform convergence of metrics $g_k$ on a closed surface withbounded integral curvature (measure) in the sense of A.D. Alexandrov, under theassumption that the curvature measures $\mathbb{K}_{g_k}=\mu^1_k-\mu^2_k$,where $\mu^1_k,\mu^2_k$ are nonnegative Radon measures that converge weakly tomeasures $\mu^1,\mu^2$ respectively, and $\mu^1$ is less than $2\pi$ at eachpoint. This generalizes Yu. G. Reshetnyak's well-known result on uniformconvergence of metrics on a domain in $\mathbb{C}$, and answers affirmativelythe open question on the metric convergence on a closed surface.We also showthat a metric $g=e^{2u}g_0$ with bounded integral curvature in the sense ofAlexandrov on a closed Riemannian surface $(\Sigma,g_0)$ can be approximated bysmooth metrics in the fixed conformal class $[g_0]$ in terms of distancefunctions, curvature measures and conformal factors. Results on a closedsurface with varying conformal classes and on complete noncompact surfaces areobtained as well.

Uniform Convergence of Metrics on Alexandrov Surfaces with Bounded Integral Curvature