This self-contained treatment of measure and integration begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. From there the reader is led to the general notion of measure, to the construction of the Lebesgue integral on a measure space, and to the major limit theorems, such as the Monotone and Dominated Convergence Theorems. The treatment proceeds to $L^p$ spaces, normed linear spaces that are shown to be complete (i.e., Banach spaces) due to the limit theorems. Particular attention is paid to $L^2$ spaces as Hilbert spaces, with a useful geometrical structure. <P>Having gotten quickly to the heart of the matter, the text proceeds to broaden its scope. There are further constructions of measures, including Lebesgue measure on $n$-dimensional Euclidean space. There are also discussions of surface measure, and more generally of Riemannian manifolds and the measures they inherit, and an appendix on the integration of differential forms. Further geometric aspects are explored in a chapter on Hausdorff measure. The text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales. <P>This text will prepare graduate students for more advanced studies in functional analysis, harmonic analysis, stochastic analysis, and geometric measure theory.
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"Taylor's treatment throughout is elegant and very efficient ... I found reading the text very enjoyable." ---- MAA Reviews
"The book is very understandable, requiring only a basic knowledge of analysis. It can be warmly recommended to a broad spectrum of readers, to graduate students as well as young researchers." ---- EMS Newsletter
"This monograph provides a quite comprehensive presentation of measure and integration theory and of some of their applications." ---- Mathematical Reviews
"The book is very understandable, requiring only a basic knowledge of analysis. It can be warmly recommended to a broad spectrum of readers, to graduate students as well as young researchers." ---- EMS Newsletter
"This monograph provides a quite comprehensive presentation of measure and integration theory and of some of their applications." ---- Mathematical Reviews
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Inside This Book
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Key Phrases - Statistically Improbable Phrases (SIPs):
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maximal function estimate, metric outer measure, covering lemma, countable cover, middle third set, countable subadditivity, mean ergodic theorem, finite signed measures, disjoint balls, monotone convergence theorem, positive linear functional, monotone class, compact metric space, maximal inequality, finite disjoint unions Key Phrases - Capitalized Phrases (CAPs): (learn more)
Dominated Convergence Theorem, Fubini's Theorem, Vitali Covering Lemma, Fundamental Theorem of Calculus, Markov Property, Stone-Weierstrass Theorem, Lusin's Theorem, Monotone Class Lemma, Radon-Nikodyrn Theorem, Tychonov's Theorem, Zorn's Lemma New!
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maximal function estimate, metric outer measure, covering lemma, countable cover, middle third set, countable subadditivity, mean ergodic theorem, finite signed measures, disjoint balls, monotone convergence theorem, positive linear functional, monotone class, compact metric space, maximal inequality, finite disjoint unions Key Phrases - Capitalized Phrases (CAPs): (learn more)
Dominated Convergence Theorem, Fubini's Theorem, Vitali Covering Lemma, Fundamental Theorem of Calculus, Markov Property, Stone-Weierstrass Theorem, Lusin's Theorem, Monotone Class Lemma, Radon-Nikodyrn Theorem, Tychonov's Theorem, Zorn's Lemma New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 42
books:
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1
book
cites this book:
- Ergodic Theory (Cambridge Studies in Advanced Mathematics) by Karl E. Petersen in Back Matter (1), and Back Matter (2)
- Advanced Calculus, Revised Edition by Loomis in Back Matter
- Lectures on Differential Geometry by Sholomo Sternberg in Back Matter
- Complex Analysis (Princeton Lectures in Analysis) by Elias M. Stein in Back Matter
- Stochastic Integrals (Ams Chelsea Publishing) by Henry P. McKean in Back Matter
See all 42 books this book cites
- Mathematical Principles of Signal Processing by Pierre Bremaud on page 261
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