Hartman / Mikusinski / Sneddon The Theory of Lebesgue Measure and Integration

1. Auflage 2014
ISBN: 978-1-4832-8033-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 15, 176 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

The Theory of Lebesgue Measure and Integration deals with the theory of Lebesgue measure and integration and introduces the reader to the theory of real functions. The subject matter comprises concepts and theorems that are now considered classical, including the Yegorov, Vitali, and Fubini theorems. The Lebesgue measure of linear sets is discussed, along with measurable functions and the definite Lebesgue integral. Comprised of 13 chapters, this volume begins with an overview of basic concepts such as set theory, the denumerability and non-denumerability of sets, and open sets and closed sets on the real line. The discussion then turns to the theory of Lebesgue measure of linear sets based on the method of M. Riesz, together with the fundamental properties of measurable functions. The Lebesgue integral is considered for both bounded functions - upper and lower integrals - and unbounded functions. Later chapters cover such topics as the Yegorov, Vitali, and Fubini theorems; convergence in measure and equi-integrability; integration and differentiation; and absolutely continuous functions. Multiple integrals and the Stieltjes integral are also examined. This book will be of interest to mathematicians and students taking pure and applied mathematics.

Autoren/Hrsg.

Hartman, S.

Mikusinski, J.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;The Theory of Lebesgue Measure and Integration;4
3;Copyright Page;5
4;Table of Contents;6
5;Foreword to the English Edition;8
6;CHAPTER I. INTRODUCTORY CONCEPTS;10
6.1;1. Sets;10
6.2;2. Denumerability and nondenumerability;12
6.3;3. Open sets and closed sets on the real line;15
7;CHAPTER
II. LEBESGUE MEASURE OF LINEAR SETS;19
7.1;1. Measure of open sets;19
7.2;2. Definition of Lebesgue measure. Measurability;21
7.3;3. Countable additivity of measure;25
7.4;4. Sets of measure zero;29
7.5;5. Non-measurable sets;31
8;CHAPTER
III. MEASURABLE FUNCTIONS;34
8.1;1. Measurability of functions;34
8.2;2. Operations on measurable functions;36
8.3;3. Addenda;40
9;CHAPTER
IV. THE DEFINITE LEBESGUE INTEGRAL;43
9.1;1. The integral of a bounded function;43
9.2;2. Generalization to unbounded functions;53
9.3;3. Integration of sequences of functions;61
9.4;4. Comparison of the Riemann and Lebesgue integrals;65
9.5;5. The integral on an infinite interval;67
10;CHAPTER
V. CONVERGENCE IN MEASURE AND EQUI-INTEGRABILITY;71
10.1;1. Convergence in measure;71
10.2;2. Equi-integrability;80
11;CHAPTER
VI. INTEGRATION AND DIFFERENTIATION FUNCTIONS OF FINITE VARIATION;84
11.1;1. Preliminary remarks;84
11.2;2. Functions of finite variation;86
11.3;3. The derivative of an integral;95
11.4;4. Density points;99
12;CHAPTER
VII. ABSOLUTELY CONTINUOUS FUNCTIONS;100
12.1;1. Definition and fundamental properties;100
12.2;2. The approximation of measurable functions by continuous
functions;105
13;CHAPTER
VIII. SPACES OF p-th POWER INTEGRABLE FUNCTIONS;108
13.1;1. The classes Lp(a,b);108
13.2;2. Arithmetic and geometric means;108
13.3;3. Holder's inequality;109
13.4;4. Minkowski's inequality;110
13.5;5. The classes Lp considered as metric spaces;111
13.6;6. Mean convergence of order p;112
13.7;7. Approximation by continuous functions;115
14;CHAPTER
IX. ORTHOGONAL EXPANSIONS;120
14.1;1. General properties;120
14.2;2. Completeness;125
15;CHAPTER
X. COMPLEX-VALUED FUNCTIONS OF A REAL VARIABLE;131
15.1;1. The Holder and Minkowski inequalities for p, q < 1;131
15.2;2. Integrals of complex-valued functions;133
15.3;3. The expansion of complex-valued functions in orthogonal
series;134
16;CHAPTER
XI. MEASURE IN THE PLANE AND IN SPACE;137
16.1;1. Definition and properties;137
16.2;2. Plane measure and linear measure;144
17;CHAPTER
XII. MULTIPLE INTEGRALS;149
17.1;1. Definition and fundamental properties;149
17.2;2. Multiple integrals and iterated integrals;151
17.3;3. The double integral on unbounded sets;157
17.4;4. Applications;160
18;CHAPTER
XIII. THE STIELTJES INTEGRAL;164
18.1;1. Definition and existence;164
18.2;2. Integration by parts and the limit of integrals;166
18.3;3. Relation between the Stielt j es integral and Lebesgue
integral;168
19;Literature;172
20;Index;176