Ebook: Measure Theory

This book was planned originally not as a work to be published, but as an excuse to buy a computer, incidentally to give me a chance to organize my own ideas ~n what measure theory every would-be analyst should learn, and to detail my approach to the subject. When it turned out that Springer-Verlag thought that the point of view in the book had general interest and offered to publish it, I was forced to try to write more clearly and search for errors. The search was productive. Readers will observe the stress on the following points. The application of pseudometric spaces. Pseudo metric, rather than metric spaces, are applied to obviate the artificial replacement of functions by equivalence classes, a replacement that makes the use of "almost everywhere" either improper or artificial. The words "function" and "the set on which a function has values at least E" can be taken literally in this book. Pseudometric space properties are applied in many contexts. For example, outer measures are used to pseudometrize classes of sets and the extension of a finite measure from an algebra to a 0" algebra is thereby reduced to finding the closure of a subset of a pseudo metric space.

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This book is different from other books on measure theory in that it accepts probability theory as an essential part of measure theory. This means that many examples are taken from probability; that probabilistic concepts such as independence, Markov processes, and conditional expectations are integrated into the text rather than being relegate to an appendix; that more attention is paid to the role of algebras than is customary; and that the metric defining the distance between sets as the measure of their symmetric difference is exploited more than is customary.

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This book is different from other books on measure theory in that it accepts probability theory as an essential part of measure theory. This means that many examples are taken from probability; that probabilistic concepts such as independence, Markov processes, and conditional expectations are integrated into the text rather than being relegate to an appendix; that more attention is paid to the role of algebras than is customary; and that the metric defining the distance between sets as the measure of their symmetric difference is exploited more than is customary.
Content:
Front Matter....Pages i-xii
Conventions and Notation....Pages 1-5
Operations on Sets....Pages 7-10
Classes of Subsets of a Space....Pages 11-16
Set Functions....Pages 17-36
Measure Spaces....Pages 37-52
Measurable Functions....Pages 53-72
Integration....Pages 73-101
Hilbert Space....Pages 103-121
Convergence of Measure Sequences....Pages 123-143
Signed Measures....Pages 145-156
Measures and Functions of Bounded Variation on R ....Pages 157-177
Conditional Expectation; Martingale Theory....Pages 179-204
Back Matter....Pages 205-212

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This book is different from other books on measure theory in that it accepts probability theory as an essential part of measure theory. This means that many examples are taken from probability; that probabilistic concepts such as independence, Markov processes, and conditional expectations are integrated into the text rather than being relegate to an appendix; that more attention is paid to the role of algebras than is customary; and that the metric defining the distance between sets as the measure of their symmetric difference is exploited more than is customary.
Content:
Front Matter....Pages i-xii
Conventions and Notation....Pages 1-5
Operations on Sets....Pages 7-10
Classes of Subsets of a Space....Pages 11-16
Set Functions....Pages 17-36
Measure Spaces....Pages 37-52
Measurable Functions....Pages 53-72
Integration....Pages 73-101
Hilbert Space....Pages 103-121
Convergence of Measure Sequences....Pages 123-143
Signed Measures....Pages 145-156
Measures and Functions of Bounded Variation on R ....Pages 157-177
Conditional Expectation; Martingale Theory....Pages 179-204
Back Matter....Pages 205-212
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