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

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 ${\mathrm{L}}^1$ of a certain sequence of step functions, and we prove it using only basic elements from Lebesgue integration theory.