A integral de Lebesgue como o completamento da integral de Riemann

Abstract

This work aims to o↵er an introduction to the study of Lebesgue measure and integration. We motivate our exhibition based on two central objectives. Firstly, construct the abstract concepts of measure and integration theory in a natural and motivated way, so that the beginner reader can abstract the main ideas and central philosophy of the theory. Secondly, we give a special focus on solving problems regarding the integral, from a metric perspective. We present the results of the construction of the integral and the measure, motivated to solve the problem of the completeness of the space of integrable functions. In this sense, the results range from basic operational properties of the Lebesgue measure and integral, as well as the convergence theorems, to results involving -algebra completions, metric completions and completeness of the space of the Lebesgue integrable functions. In the end, we also show a construction that allows us to see the extension of the elementary measure to the Lebesgue measure as a metric completion.