Chapter V. Lebesgue Measure and Abstract Measure Theory

Abstract

This chapter develops the basic theory of measure and integration, including Lebesgue measure and Lebesgue integration for the line. Section 1 introduces measures, including 1-dimensional Lebesgue measure as the primary example, and develops simple properties of them. Sections 2–4 introduce measurable functions and the Lebesgue integral and go on to establish some easy properties of integration and the fundamental theorems about how Lebesgue integration behaves under limit operations. Sections 5–6 concern the Extension Theorem announced in Section 1 and used as the final step in the construction of Lebesgue measure. The theorem allows $\sigma$-finite measures to be extended from algebras of sets to $\sigma$-algebras. The theorem is proved in Section 5, and the completion of a measure space is defined in Section 6 and related to the proof of the Extension Theorem. Section 7 treats Fubini's Theorem, which allows interchange of order of integration under rather general circumstances. This is a deep result. As part of the proof, product measure is constructed and important measurability conditions are established. This section mentions that Fubini's Theorem will be applicable to higher-dimensional Lebesgue measure, but the details are deferred to Chapter VI. Section 8 extends Lebesgue integration to complex-valued functions and to functions with values in finite-dimensional vector spaces. Section 9 gives a careful definition of the spaces $L^1$, $L^2$, and $L^{\infty}$ for any measure space, introduces the notion of a normed linear space, and verifies that these three spaces are examples. The main theorem of the section about $L^1$, $L^2$, and $L^{\infty}$ is the completeness of these three spaces as metric spaces. In addition, the section proves a version of Alaoglu's Theorem concerning weak-star convergence.