Measure and Integral
Marcel Dekker Inc (Verlag)
978-0-8247-6499-9 (ISBN)
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This volume develops the classical theory of the Lebesgue integral and some of its applications. The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given.
Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis--harmonic analysis--are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function.
Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas.
Preliminaries
Points and Sets in Rn
Rn as a Metric Space
Open and Closed Sets in Rn: Special Sets
Compact Sets; The Heine-Borel Theorem
Functions
Continuous Functions and Transformations
The Riemann Integral
Exercises Function of Bounded Variation; The Riemann-Stieltjes Integral Functions of Bounded Variation
Rectifiable Curves
The Reiman-Stieltjes Integral
Further Results About the Reimann-Stieltjes Integrals
Exercises
Lebesgue Measure and Outer Measure Lebesgue Outer Measures; The Cantor Set. Lebesgue Measurable Sets
Two Properties of Lebesgue Measure
Characterizations of Measurability
Lipschitz Transformations of Rn
A Nonmeasurable Set. Exercises
Lebesgue Measurable Functions
Elementary Properties of Measurable Functions. Semicontinuous Functions
Properties of Measurable Functions; Egorov's Theorem and Lusin's Theorem
Convergence in Measure
Exercises
The Lebesgue Integral
Definition of the Integral of a Nonnegative Function
Properties of the Integral
The Integral of an Arbitrary Measurable f
A Relation Between Riemann-Stieltjes and Lebesgue Integrals; the LP Spaces, 0
| Erscheint lt. Verlag | 1.11.1977 |
|---|---|
| Reihe/Serie | Chapman & Hall/CRC Pure and Applied Mathematics |
| Verlagsort | New York |
| Sprache | englisch |
| Maße | 152 x 229 mm |
| Gewicht | 544 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 0-8247-6499-4 / 0824764994 |
| ISBN-13 | 978-0-8247-6499-9 / 9780824764999 |
| Zustand | Neuware |
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