Littlewood's fourth principle

TL;DR

This paper discusses Littlewood's four principles in measure theory, proposing a new principle that simplifies proofs of key theorems and highlights the analogy between topological and measure-theoretic concepts.

Contribution

It introduces a fourth principle in measure theory that simplifies proofs of classical theorems and emphasizes the analogy between topology and measure theory.

Findings

Simplified proofs of Lusin's and Egoroff-Severini's theorems

Egoroff-Severini's theorem as a natural extension of Dini's theorem

Enhanced understanding of measure and topology analogies

Abstract

In Real Analysis, Littlewood's three principles are known as heuristics that help teach the essentials of measure theory and reveal the analogies between the concepts of topological space and continuos function on one side and those of measurable space and measurable function on the other one. They are based on important and rigorous statements, such as Lusin's and Egoroff-Severini's theorems, and have ingenious and elegant proofs. We shall comment on those theorems and show how their proofs can possibly be made simpler by introducing a \textit{fourth principle}. These alternative proofs make even more manifest those analogies and show that Egoroff-Severini's theorem can be considered the natural generalization of the classical Dini's monotone convergence theorem.