Let γ(E) be the analytic capacity of a compact set E and let
γ+(E) be the capacity of E originated by Cauchy transforms of
positive measures. In this paper we prove that γ(E)≈γ+(E)
with estimates independent of E. As a corollary, we characterize removable
singularities for bounded analytic functions in terms of curvature of measures,
and we deduce that γ is semiadditive, which solves a long standing
question of Vitushkin.Comment: 42 page