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 $K_{g_k}={\mu }_k^1-{\mu }_k^2$, where ${\mu }_k^1,{\mu }_k^2$ are nonnegative Radon measures converging weakly to measures ${\mu }^1,{\mu }^2$ respectively, and ${\mu }^1$ is less than 2π at each point (no cusps). This is the global version of 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. We also give an analytic proof of the fact that a (singular) metric $g=e^{2u}{g}_0$ with bounded integral curvature on a closed Riemannian surface $(\Sigma ,g_0)$ can be approximated by smooth metrics in the fixed conformal class $\left[g_0\right]$. Results on a closed surface with varying conformal classes and on complete noncompact surfaces are obtained as well.