Chapter 4 - Special Topics of Lebesgue Integral and Applications

Abstract

This chapter discusses the relationship between differentiation and integration on a set of real numbers, R. It explores the conditions under which the fundamental theorem of calculus is valid. This exploration leads to interesting and unexpected measure-theoretic ideas. The main result in this direction is the Lebesgue-Radon-Nikodym theorem. The chapter also presents some of the main theorems in probability closely related to measure theory and Lebesgue integration. According to Lebesgue's theorem, a monotone function has a finite derivative almost everywhere. The main tool used in the proof is the Vitali's Covering Theorem. Vitali's theorem has many applications in classical analysis, particularly in the theory of differentiation. If f is the difference of two monotone functions, then f is differentiable almost everywhere. Differentiation and integration are inverse operations to each other on the space of smooth functions. It has been widely accepted that probabilities should be studied as special sorts of measures. In the study of probability distributions, the center point and the overall spread of information of the distribution are of great interest. These usually can be obtained by computing the mean value and standard deviation.