Real Analysis

Measure Theory, Integration, and Hilbert Spaces

Elias M. Stein, Rami Shakarchi

• Publisher: Princeton University Press
• Year: 2009
• ISBN:
  9781400835560
• Topic: Matematica
• Subtopic: Analisi matematica

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.
After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises.
As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.
Also available, the first two volumes in the Princeton Lectures in Analysis:

1 Measure Theory

The sets whose measure we can define by virtue of the preceding ideas we will call measurable sets; we do this without intending to imply that it is not possible to assign a measure to other sets.

E. Borel, 1898

This chapter is devoted to the construction of Lebesgue measure in

^d and the study of the resulting class of measurable functions. After some preliminaries we pass to the first important definition, that of exterior measure for any subset E of

^d. This is given in terms of approximations by unions of cubes that cover E. With this notion in hand we can define measurability and thus restrict consideration to those sets that are measurable. We then turn to the fundamental result: the collection of measurable sets is closed under complements and countable unions, and the measure is additive if the subsets in the union are disjoint.

The concept of measurable functions is a natural outgrowth of the idea of measurable sets. It stands in the same relation as the concept of continuous functions does to open (or closed) sets. But it has the important advantage that the class of measurable functions is closed under pointwise limits.

1 Preliminaries

We begin by discussing some elementary concepts which are basic to the theory developed below.

The main idea in calculating the “volume” or “measure” of a subset of

^d consists of approximating this set by unions of other sets whose geometry is simple and whose volumes are known. It is convenient to speak of “volume” when referring to sets in

^d; but in reality it means “area” in the case d = 2 and “length” in the case d = 1. In the approach given here we shall use rectangles and cubes as the main building blocks of the theory: in

we use intervals, while i...