Lebesgue integration : a new proof of the fundamental theorem of calculus and a new lebesgue integrable function

Abstract

In this thesis, we present a new proof of the fundamental theorem of calculus of the Lebesgue Integral [1] and we explore the properties of a new and interesting Lebesgue integrable function [2]. In this thesis, we reveal the main connection between “absolute continuity” and “Lebesgue Integration” that is established from the fundamental theorem of calculus for the Lebesgue integral. And a function which is Lebesgue integrable but not Riemann integrable. In chapter 1, an elementary and simple proof of the fundamental theorem calculus for the Lebesgue integral is given. The proof is based on the construction of a sequence of step functions that converges in L¹. In chapter 2, we study the properties of a new function which is Lebesgue integrable but not Riemann integrable.