Uma introdução à Teoria da Medida e Integração no Rn

Abstract

lts of Integration and Measure theory in R n . We begin with the basic definitions of rectangles and cubes. Then, we introduce the concepts of exterior measure, measurable sets, Lebesgue measure, and the Borel sigma-algebra. After, we focus on measurable functions and the Lebesgue integral, detailing some of their fundamental properties. In the second part, we study some important results in the theory of integration, such as the theorem of approximation by simple functions, the three Littlewood Principles, Fatou’s Lemma, the Monotone convergence theorem, and the Dominated convergence theorem. We also prove the known theorems of Fubini and Tonelli, which are crucial for extending the integration theory to multiple dimensions. Finally, we investigate the relationship between integration and differentiation in this new context of measurable functions and present the important Lebesgue differentiation theorem. We conclude presenting a version of the Fundamental theorem of calculus applied to measurable functions.