Extension of the Continuity Theorems of Lebesgue Integration

Abstract

The Dominated Convergence Theorem, Fatou's Lemma, and the Monotone Convergence Theorem are collectively referred to as the continuity theorems of Lebesgue integration. These ubiquitous results are stated and proved in the literature for sequences of measurable functions. It is well known that these results do not hold for arbitrary nets of measurable functions. These results, as well as the Levy Continuity Theorem, Riesz's characterization of $\mathcal{L}_p$ convergence in terms of almost everywhere convergence and convergence of norms, and a stronger version of Scheffe's Theorem, are derived for nets indexed by countably accessible directed sets.