A simple proof of the Lebesgue decomposition theorem

TL;DR

This paper provides an elementary, self-contained, and direct proof of the classical Lebesgue decomposition theorem, making the proof more accessible and straightforward for learners and researchers.

Contribution

It introduces a simplified and self-contained proof of the Lebesgue decomposition theorem, enhancing understanding and accessibility.

Findings

Provides an elementary proof accessible to a broader audience

Simplifies the understanding of the Lebesgue decomposition theorem

Maintains rigor while reducing complexity

Abstract

The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.