A simple proof of the Fundamental Theorem of Calculus for the Lebesgue integral

TL;DR

This paper presents an elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral, focusing on convergence in L^1 of step functions using basic Lebesgue theory.

Contribution

It provides a new, simple proof of the theorem, emphasizing elementary methods and convergence in L^1.

Findings

Elementary proof of the Fundamental Theorem of Calculus for Lebesgue integral

Convergence in L^1 of step functions established using basic Lebesgue theory

Simplifies understanding of the theorem's proof

Abstract

This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in ${\rm L}^1$ of a certain sequence of step functions, and we prove it using only basic elements from Lebesgue integration theory.