Measures and Integration: An Informal Introduction

Abstract

For many students who are learning measure and integration theory for the first time, the notions of a σ-algebra of subsets of a set Ω, countable additivity of a set function λ, measurability of a function, the definition of an integral, and the interchange of limits and integration are not easy to understand and often seem not so intuitive. The goals of this informal introduction to this subject are (1) to show that the notions of σ-algebra and countable additivity are logical consequences of certain natural ap-proximation procedures; (2) the dividends for the assumption of these two properties are great, and they lead to a nice and natural theory that is also very powerful for the handling of limits. Of course, as the saying goes, the devil is in the details. After this informal introduction, the necessary de-tails are given in the next few sections. It is hoped that after this heuristic explanation of the subject, the motivation for and the process of mastering the details on the part of the students will be forthcoming. What is Measure Theory? A simple answer is that it is a theory about the distribution of mass over a set S. If the mass is uniformly distributed and S is an Euclidean space R k , it is the theory of Lebesgue measure on R k (i.e., length in R, area in R 2 , volume in R 3 , etc.). Probability theory is concerned with the case when S is the sample space of a random experiment and the total mass is one. Consider the following example. Imagine an open field S and a snowy night. At daybreak one goes to the field to measure the amount of snow in as many of the subsets of S as Excerpt from " Measure Theory and Probability Theory, " by Krishna B. Athreya and Soumendra N. Lahiri , Springer (2006).