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Written by one of the leading scholars in the field
Includes a novel presentation of differentiation and absolute continuity using a local maximum function, resulting in an exposition that is both simpler and more general than the traditional approach
Theorems are stated for Lebesgue and Borel measures, with a note indicating when the same proof works only for Lebesgue measures
Appendices cover additional material, including theorems for higher dimensions and a short introduction to nonstandard analysis
Includes supplementary material: sn.pub/extras
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Table of contents (11 chapters)
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Front Matter
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Back Matter
About this book
The first half of the book develops both Lebesgue measure and, with essentially no additional work for the student, general Borel measures for the real line. Notation indicates when a result holds only for Lebesgue measure. Differentiation and absolute continuity are presented using a local maximal function, resulting in an exposition that is both simpler and more general than the traditional approach.
The second half deals with general measures and functional analysis, including Hilbert spaces, Fourier series, and the Riesz representation theorem for positive linear functionals on continuous functions with compact support. To correctly discuss weak limits of measures, one needs the notion of a topological space rather than just a metric space, so general topology is introduced in terms of a base of neighborhoods at a point. The development of results then proceeds in parallel with results for metric spaces, where the base is generated by balls centered at a point. The text concludes with appendices on covering theorems for higher dimensions and a short introduction to nonstandard analysis including important applications to probability theory and mathematical economics.
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Authors and Affiliations
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University of Illinois, Urbana, USA
Peter A Loeb
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Bibliographic Information
Book Title: Real Analysis
Authors: Peter A Loeb
DOI: https://doi.org/10.1007/978-3-319-30744-2
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2016
Hardcover ISBN: 978-3-319-30742-8Published: 16 May 2016
Softcover ISBN: 978-3-319-80879-6Published: 27 May 2018
eBook ISBN: 978-3-319-30744-2Published: 05 May 2016
Edition Number: 1
Number of Pages: XII, 274
Topics: Real Functions, Functional Analysis, Measure and Integration