The theory of Lebesgue measure and integration [electronic resource] / by S. Hartman and J. Mikusiński.
The Theory of Lebesgue Measure and Integration.
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Full Text (via ScienceDirect) |
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| Format: | Electronic eBook |
| Language: | English Polish |
| Published: |
New York :
Pergamon Press,
[1961]
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| Edition: | Enl. ed., |
| Series: | International series of monographs in pure and applied mathematics ;
v. 15. |
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Table of Contents:
- Front Cover; The Theory of Lebesgue Measure and Integration; Copyright Page; Table of Contents; Foreword to the English Edition; CHAPTER I. INTRODUCTORY CONCEPTS; 1. Sets; 2. Denumerability and nondenumerability; 3. Open sets and closed sets on the real line; CHAPTERII. LEBESGUE MEASURE OF LINEAR SETS; 1. Measure of open sets; 2. Definition of Lebesgue measure. Measurability; 3. Countable additivity of measure; 4. Sets of measure zero; 5. Non-measurable sets; CHAPTERIII. MEASURABLE FUNCTIONS; 1. Measurability of functions; 2. Operations on measurable functions; 3. Addenda.
- CHAPTERIV. THE DEFINITE LEBESGUE INTEGRAL1. The integral of a bounded function; 2. Generalization to unbounded functions; 3. Integration of sequences of functions; 4. Comparison of the Riemann and Lebesgue integrals; 5. The integral on an infinite interval; CHAPTERV. CONVERGENCE IN MEASURE AND EQUI-INTEGRABILITY; 1. Convergence in measure; 2. Equi-integrability; CHAPTERVI. INTEGRATION AND DIFFERENTIATION FUNCTIONS OF FINITE VARIATION; 1. Preliminary remarks; 2. Functions of finite variation; 3. The derivative of an integral; 4. Density points; CHAPTERVII. ABSOLUTELY CONTINUOUS FUNCTIONS.
- 1. Definition and fundamental properties2. The approximation of measurable functions by continuousfunctions; CHAPTERVIII. SPACES OF p-th POWER INTEGRABLE FUNCTIONS; 1. The classes Lp(a, b); 2. Arithmetic and geometric means; 3. Holder's inequality; 4. Minkowski's inequality; 5. The classes Lp considered as metric spaces; 6. Mean convergence of order p; 7. Approximation by continuous functions; CHAPTERIX. ORTHOGONAL EXPANSIONS; 1. General properties; 2. Completeness; CHAPTERX. COMPLEX-VALUED FUNCTIONS OF A REAL VARIABLE; 1. The Holder and Minkowski inequalities for p, q <1.
- 2. Integrals of complex-valued functions3. The expansion of complex-valued functions in orthogonalseries; CHAPTERXI. MEASURE IN THE PLANE AND IN SPACE; 1. Definition and properties; 2. Plane measure and linear measure; CHAPTERXII. MULTIPLE INTEGRALS; 1. Definition and fundamental properties; 2. Multiple integrals and iterated integrals; 3. The double integral on unbounded sets; 4. Applications; CHAPTERXIII. THE STIELTJES INTEGRAL; 1. Definition and existence; 2. Integration by parts and the limit of integrals; 3. Relation between the Stielt j es integral and Lebesgueintegral; Literature.