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MEASURE, INTEGRATION AND FUNCTION SPACES
by Charles Swartz (New Mexico State University)
This text contains a basic introduction to the abstract measure theory and the Lebesgue integral. Most of the standard topics in the measure and integration theory are discussed. In addition, topics on the Hewitt-Yosida decomposition, the Nikodym and Vitali-Hahn-Saks theorems and material on finitely additive set functions not contained in standard texts are explored. There is an introductory section on functional analysis, including the three basic principles, which is used to discuss many of the classic Banach spaces of functions and their duals. There is also a chapter on Hilbert space and the Fourier transform.
Contents:
- Measure Theory:
- Lebesgue Measure
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Lebesgue-Stieltjes Measure
- The Nikodym Convergence and Boundedness Theorems
- Integration:
- Measurable Functions
- The Lebesgue Integral Convergence in Mean
- Convergence in Measure
- Comparison of Modes of Convergence
- The Radon-Nikodym Theorem
- The Vitali-Hahn-Saks Theorem
- Differentiation and Integration:
- Integrating Derivatives
- Absolutely Continuous Functions
- Introduction to Functional Analysis:
- Normed Linear Spaces (NLS)
- Quotient Spaces
- The Hahn-Banach Theorem
- Ordered Linear Spaces
- Function Spaces:
- Lp-Spaces, 1 £ p < ¥
- The Space of Finitely Additive Set Functions
- The Space of Countably Additive Set Functions
- The Space of Continuous Functions
- Hilbert Space
- and other papers
Readership: Mathematicians and graduate students in mathematics.
"... this monograph can be recommended to be read ... as a source of inspiration."
"... the author has provided us with a very rich and attractive treatment with an unusually extensive coverage of the area ... One particularly valuable aspect of the book is that it gives a much richer discussion of additive set functions than is usual. Another is that it contains many important theorems (such as those of Drewnowski, Mikusinski, Nikodym, Vitali-Hahn-Saks, Yosida-Hewitt, etc.) that are not usually included in texts."
"This excellent text is recommended to students and teachers interested in the measure and integration theory ... More than 70 examples and 400 exercises make the volume valuable, enjoyable and very useful."
| 292pp |
Pub. date: Feb 1994 |
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