Convergence from below suffices

TL;DR

The paper presents a strengthened version of the monotone convergence theorem, called the convergence from below theorem, derived using Fatou's lemma, and advocates for its inclusion in introductory measure and integration courses.

Contribution

It introduces and promotes the convergence from below theorem as a fundamental and accessible result in measure theory, derived simply from Fatou's lemma.

Findings

Provides a clearer understanding of convergence from below

Demonstrates the theorem's simplicity and educational value

Suggests wider recognition in measure theory education

Abstract

An elementary application of Fatou's lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem. We make the case that this result should be better known, and deserves a place in any introductory course on measure and integration.