Lebesgue Measure

Abstract

The most common approach to Lebesgue integration is to start with the notion of measure, which extends the simple idea of length to more complicated sets, and use it to define measurable functions and their Lebesgue integral. We have chosen Riesz’s approach and have introduced the Lebesgue integral first. Therefore, we shall define the idea of measure using the integral. As we have seen, the space L¹(ℝ) of all Lebesgue integrable functions studied in Chapter 10 is a large class of functions. Unfortunately, it does not contain some of the nicest functions we constantly encounter in analysis: For instance, it does not contain continuous functions or even constant functions on unbounded intervals. Also, although convergence theorems such as Lebesgue’s Dominated Convergence Theorem are valid under very reasonable conditions, a sequence of integrable functions that converges almost everywhere does not, in general, have an integrable limit. We are therefore interested in a larger class of functions containing simultaneously all functions one encounters in practice. One of our goals in this chapter will be to introduce and study this class. Although we start with F. Riesz’s definition of a measurable function, we shall later give the more general definitions of measure, measurable sets, and measurable functions and prove the equivalence of the corresponding definitions for Lebsegue measure. In addition, we shall give another definition of the Lebesgue integral for measurable functions and show that it is equivalent to Riesz’s definition given in the previous chapter. Throughout this chapter, I, J, etc. will denote (possibly unbounded) intervals of ℝ.