The Fundamental Convergence Theorem and the Reduction Operation

Abstract

Theorem. Let Г: $\left\{ {{{u}_{\alpha }},\alpha \in I} \right\}$ be a family of superharmonic functions defined on an open subset of ℝ^N, locally uniformly bounded below, and define the lower envelope u by u(ξ) = inf$_{{\alpha \in I}}{{u}_{\alpha }}(\xi )$.