Lebesgue Measure and Integration

We shall begin our study of Lebesgue measure and integration theory from some fundamental, general no-

tions.

The concept of measure of a set arises from the problem of generalizing the notion of “size” of sets in IR

and IRn and extending such notions to arbitrary sets. Thus, the measure of a set A = (a, b) ⊂ IR is merely its length, the measure of a set A ⊂ IR2 is its area, and of a set A ⊂ IR3, its volume. In more general situations, the idea of the size of a set is less clear. Measure theory is the mathematical theory concerned

with these generalizations and is an indispensable part of functional analysis. The benefits of generalizing

ideas of size of sets are substantial, and include the development of a rich and powerful theory of integration

that extends and generalizes elementary Riemann integration outlined in Chapter 1. Now, we find that the

basic mathematical properties of sizes of geometrical objects, such as area and volume, are shared by other

types of sets of interest, such as sets of random events and the probability of events taking place. Our plan

here is to give a brief introduction to this collection of ideas, which includes the ideas of Lebesgue measure

and integration essential in understanding fundamental examples of metric and normed spaces dealt with in

subsequent chapters. We begin with the concept of σ-algebra.