  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 $\mathbb{K}_{g_k}=\mu^1_k-\mu^2_k$,
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 $\mathbb{C}$, and answers affirmatively
the open question on the metric convergence on a closed surface.We also show
that a metric $g=e^{2u}g_0$ with bounded integral curvature in the sense of
Alexandrov on a closed Riemannian surface $(\Sigma,g_0)$ can be approximated by
smooth metrics in the fixed conformal class $[g_0]$ in terms of distance
functions, curvature measures and conformal factors. Results on a closed
surface with varying conformal classes and on complete noncompact surfaces are
obtained as well.
