A converse to Lebesgue's dominated convergence theorem

Abstract

Let (X, B, m) be a measure space and let f(x) be a real-valued or complex-valued measurable function on X. A non-negative measurable function s(x) will be said to dominate f(x) provided |f(x)| ≦ s(x) for almost all x in X. The function s(x) will be said to dominate the sequence {f(x)}n∈N, N = {1, 2,…}, provided it dominates each fn(x) in the sequence. Unless otherwise specified, each integral will be over X with respect to m.