A concise introduction to the theory of integration

Daniel W. Stroock9780817637590, 0817637591, 3764337591

Designed for the full-time analyst, physicists, engineer, or economist, this book attempts to provide its readers with most of the measure theory they will ever need. Given the choice, the author has consistently opted to develop the concrete rather than the abstract aspects of topics treated.
The major new feature of this third edition is the inclusion of a new chapter in which the author introduces the Fourier transform. In that Hermite functions play a central role in his treatment of Parseval’s identity and the inversion formula, Stroock’s approach bears greater resemblance to that adopted by Norbert Wiener than that used in most modern introductory texts. A second feature is that solutions to all problems are provided.
As a self-contained text, this book is excellent for both self-study and the classroom.

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Table of contents :
Contents ……Page 4
Preface to the First Edition ……Page 6
Preface to the Second Edition ……Page 7
1.1 Riemann Integration ……Page 8
1.2 Riemann-Stieltjes Integration ……Page 14
2.0 The Idea ……Page 26
2.1 Existence ……Page 28
2.2 Euclidean Invariance ……Page 37
3.1 Measure Spaces ……Page 41
3.2 Construction of Integrals ……Page 47
3.3 Convergence of Integrals ……Page 57
3.4 Lebesgue’s Differentiation Theorem ……Page 69
4.1 Fubini’s Theorem ……Page 75
4.2 Steiner Symmetrization and the Isodiametric Inequality ……Page 81
5.0 Introduction ……Page 87
5.1 Lebesgue Integrals vs. Riemann Integrals ……Page 88
5.2 Polar Coordinates ……Page 92
5.3 Jacobi’s Transformation and Surface Measure ……Page 96
5.4 The Divergence Theorem ……Page 110
6.1 Jensen, Minkowski, and Holder ……Page 121
6.2 The Lebesgue Spaces ……Page 126
6.3 Convolution and Approximate Identities ……Page 138
7.1 An Existence Theorem ……Page 146
7.2 Hubert Space and the Radon-Nikodym Theorem ……Page 158
Notation ……Page 166
Index ……Page 169
Solution to Selected Problems ……Page 173