Um estudo sobre os espaços Lp

Abstract

This work aims to study the spaces of Lebesgue integrable functions, verifying their characteristics and properties. One of the most important results of this work is the Riesz-Fischer Theorem which shows us that the p function space integrable to Lebesgue is a Banach space. For the construction of these spaces was made the study of the Lebesgue integral, which is nothing more than a generalization of the Riemann integral. A study on set algebra was made and thus defined what are algebras and sigma-algebras, and also verified which is the largest sigma-algebra of subsets of real that meets the conditions of a measure. This work will also present the main results regarding the study of measurement and integration as its main characteristics. Some results regarding the construction of measures such as a classic problem of measurement theory, exterior measures, measurable sets and measurable functions were addressed. Important convergence theorems are also presented, such as the Monotonous Convergence Theorem, Fatou's Lemma, the Lebesgue Dominated Convergence Theorem, and the Limited Convergence Theorem.