Measures, Integrals and Martingales

by René L. Schilling

• ISBN13: 9780521615259
• ISBN10: 0521615259
• Format: Paperback
• Copyright: 2006-01-16
• Publisher: Cambridge University Press

Summary

This is a concise and elementary introduction to measure and integration theory as it is nowadays needed in many parts of analysis and probability theory. The basic theory - measures, integrals, convergence theorems, Lp-spaces and multiple integrals - is explored in the first part of the book. The second part then uses the notion of martingales to develop the theory further, covering topics such as Jacobi's generalized transformation Theorem, the Radon-Nikodym theorem, differentiation of measures, Hardy-Littlewood maximal functions or general Fourier series. Undergraduate calculus and an introductory course on rigorous analysis are the only essential prerequisites, making this text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included and these are not merely drill problems but are there to consolidate what has already been learnt and to discover variants, sideways and extensions to the main material. Hints and solutions can be found on the authors website, which can be reached from www.cambridge.org/9780521615259.

Table of Contents

Prelude	viii
Dependence chart	xi
Prologue	1	(4)
Problems	4	(1)
The pleasures of counting	5	(10)
Problems	13	(2)
σ-algebras	15	(7)
Problems	20	(2)
Measures	22	(9)
Problems	28	(3)
Uniqueness of measures	31	(6)
Problems	35	(2)
Existence of measures	37	(12)
Problems	46	(3)
Measurable mappings	49	(8)
Problems	54	(3)
Measurable functions	57	(10)
Problems	65	(2)
Integration of positive functions	67	(9)
Problems	73	(3)
Integrals of measurable functions and null sets	76	(12)
Null sets and the `a.e.'	80	(4)
Problems	84	(4)
Convergence theorems and their applications	88	(17)
Parameter-dependent integrals	91	(1)
Riemann vs. Lebesgue integration	92	(6)
Examples	98	(2)
Problems	100	(5)
The function spaces Lp, 1 p ∞	105	(15)
Problems	116	(4)
Product measures and Fubini's theorem	120	(14)
More on measurable functions	127	(1)
Distribution functions	128	(2)
Minkowski's inequality for integrals	130	(1)
Problems	130	(4)
Integrals with respect to image measures	134	(8)
Convolutions	137	(3)
Problems	140	(2)
Integrals of images and Jacobi's transformation rule	142	(21)
Jacobi's transformation formula	147	(5)
Spherical coordinates and the volume of the unit ball	152	(4)
Continuous functions are dense in Lp(λn)	156	(2)
Regular measures	158	(1)
Problems	159	(4)
Uniform integrability and Vitali's convergence theorem	163	(13)
Different forms of uniform integrability	168	(5)
Problems	173	(3)
Martingales	176	(14)
Problems	188	(2)
Martingale convergence theorems	190	(12)
Problems	200	(2)
The Radon--Nikodym theorem and other applications of martingales	202	(24)
The Radon--Nikodym theorem	202	(9)
Martingale inequalities	211	(2)
The Hardy--Littlewood maximal theorem	213	(5)
Lebesgue's differentiation theorem	218	(3)
The Calderon--Zygmund lemma	221	(1)
Problems	222	(4)
Inner product spaces	226	(8)
Problems	232	(2)
Hilbert space	234	(14)
Problems	246	(2)
Conditional expectations in L2	248	(10)
On the structure of subspaces of L2	253	(4)
Problems	257	(1)
Conditional expectations in Lp	258	(18)
Classical conditional expectations	263	(6)
Separability criteria for the spaces Lp(X, A, μ)	269	(5)
Problems	274	(2)
Orthonormal systems and their convergence behaviour	276	(37)
Orthogonal polynomials	276	(7)
The trigonometric system and Fourier series	283	(6)
The Haar system	289	(6)
The Haar wavelet	295	(4)
The Rademacher functions	299	(3)
Well-behaved orthonormal systems	302	(10)
Problems	312	(1)
Appendix A: lim inf and lim sup	313	(5)
Appendix B: Some facts from point-set topology	318	(10)
Topological spaces	319	(3)
Metric spaces	322	(3)
Normed spaces	325	(3)
Appendix C: The volume of a parallelepiped	328	(2)
Appendix D: Non-measurable sets	330	(7)
Appendix E: A summary of the Riemann integral	337	(23)
The (proper) Riemann integral	337	(9)
The fundamental theorem of integral calculus	346	(5)
Integrals and limits	351	(2)
Improper Riemann integrals	353	(7)
Further reading	360	(4)
References	364	(3)
Notation index	367	(4)
Name and subject index	371