Measure Theory and Integration, Second Edition

by Rao; M.M.

• ISBN13: 9780824754013
• ISBN10: 0824754018
• Edition: 2nd
• Format: Hardcover
• Copyright: 2004-01-30
• Publisher: CRC Press

Summary

Significantly revised and expanded, this authoritative reference/text comprehensively describes concepts in measure theory, classical integration, and generalized Riemann integration of both scalar and vector types-providing a complete and detailed review of every aspect of measure and integration theory using valuable examples, exercises, and applications. With more than 170 references for further investigation of the subject, this Second Edition provides more than 60 pages of new information, as well as a new chapter on nonabsolute integrals contains extended discussions on the four basic results of Banach spaces presents an in-depth analysis of the classical integrations with many applications, including integration of nonmeasurable functions, Lebesgue spaces, and their properties details the basic properties and extensions of the Lebesgue-Carathéodory measure theory, as well as the structure and convergence of real measurable functions covers the Stone isomorphism theorem, the lifting theorem, the Daniell method of integration, and capacity theory Measure Theory and Integration, Second Edition is a valuable reference for all pure and applied mathematicians, statisticians, and mathematical analysts, and an outstanding text for all graduate students in these disciplines.

Author Biography

M. M. Rao is Professor of Mathematics, University of California, Riverside.

Table of Contents

Preface to the Second Edition	v
Preface to the First Edition	xi
1. Introduction and Preliminaries	1	(20)
1.1. Motivation and Outlook,	1	(3)
1.2. The Space Rn as a Model,	4	(10)
1.3. Abstraction of the Salient Features,	14	(7)
2. Measurability and Measures	21	(89)
2.1. Measurability and Class Properties,	21	(9)
2.2. The Lebesgue Outer Measure and the Carathéodory Process,	30	(37)
2.3. Extensions of Measures to Larger Classes,	67	(19)
2.4. Distinction between Finite and Infinite Measures,	86	(6)
2.5. Metric Outer Measures,	92	(7)
2.6. Lebesgue-Stieltjes Measures,	99	(11)
3. Measurable Functions	110	(37)
3.1. Definition and Basic Properties,	110	(10)
3.2. Measurability with Measures and Convergence,	120	(16)
3.3. Image Measures and Vague Convergence,	136	(11)
4. Classical Integration	147	(108)
4.1. The Abstract Lebesgue Integral,	147	(16)
4.2. Integration of Nonmeasurable Functions,	163	(8)
4.3. The Lebesgue Limit Theorems,	171	(20)
4.4. The Vitali-Hahn-Saks Theorem and Signed Measures,	191	(12)
4.5. The LP-spaces,	203	(35)
4.6. The Four Basic Theorems of Banach Spaces,	238	(17)
5. Differentiation and Duality	255	(109)
5.1. Variations of Set Functions and the Hahn Decomposition,	255	(13)
5.2. Absolute Continuity and Complete Monotonicity of Functions,	268	(28)
5.3. The Radon-Nikodym Theorem: Sigma-Finite Case,	296	(24)
5.4. The Radon-Nikodym Theorem: General Case,	320	(10)
5.5. Duality of LP-spaces and Conditional Expectations,	330	(34)
6. Product Measures and Integrals	364	(88)
6.1. Basic Definitions and Properties,	364	(17)
6.2. The Fubini-Stone and Tonelli Theorems,	381	(17)
6.3. Remarks on Non-Cartesian Products,	398	(7)
6.4. Infinite Product Measures,	405	(32)
6.5. Two Applications of Infinite Products,	437	(15)
7. Nonabsolute Integration	452	(111)
7.1. Nonabsolute Integration on the Line,	453	(35)
7.2. Product Spaces and P-Integration,	488	(14)
7.3. Vector Integration,	502	(15)
7.4. Boundedness Principles for Nonabsolute Integration,	517	(32)
7.5. Some Complements,	549	(14)
8. Capacity Theory and Integration	563	(36)
8.1. Preliminaries on Analytic Sets,	563	(7)
8.2. Capacity: A Construction and Choquet's Theorem,	570	(15)
8.3. Application to the Daniell Integral,	585	(14)
9. The Lifting Theorem	599	(32)
9.1. The problem, Motivation, and Preliminaries,	599	(12)
9.2. Existence Proof for the Lifting Map,	611	(11)
9.3. Topologies Induced by Lifting and Related Concepts,	622	(9)
10. Topological Measures	631	(72)
10.1. Introduction and Preliminaries,	631	(8)
10.2. Regularity of Measures,	639	(28)
10.3. Local Functionals and the Riesz-Markov Theorem,	667	(20)
10.4. Haar Measures,	687	(16)
11. Some Complements and Applications	703	(30)
11.1. Lattice and Homomorphism Properties,	703	(10)
11.2. Some Applications of the Stone Isomorphism Theorem,	713	(16)
11.3. Remarks on Topology of a Group Through Measure,	729	(4)
Appendix	733	(4)
References	737	(10)
Index of Symbols and Notation	747	(4)
Author Index	751	(4)
Subject Index	755