# The Theory of Lebesgue Measure and Integration, Volume 15

## 1st Edition

**Editors:**I. N. Sneddon M. Stark S. Ulam

**Authors:**S. Hartman J. Mikusinski

**eBook ISBN:**9781483280332

**Imprint:**Pergamon

**Published Date:**1st January 1961

**Page Count:**176

**View all volumes in this series:**International Series in Pure and Applied Mathematics

## Table of Contents

Foreword to the English Edition

I. Introductory Concepts

1. Sets

2. Denumerability and Nondenumerability

3. Open Sets and Closed Sets on the Real Line

II. 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

III. Measurable Functions

1. Measurability of Functions

2. Operations on Measurable Functions

3. Addenda

IV. The Definite Lebesgue Integral

1. 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

V. Convergence in Measure and Equi-Integrability

1. Convergence in Measure

2. Equi-Integrability

VI. Integration and Differentiation. Functions of Finite Variation

1. Preliminary Remarks

2. Functions of Finite Variation

3. The Derivative of an Integral

4. Density Points

VII. Absolutely Continuous Functions

1. Definition and Fundamental Properties

2. The Approximation of Measurable Functions by Continuous Functions

VIII. 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

IX. Orthogonal Expansions

1. General Properties

2. Completeness

X. Complex-Valued Functions of a Real Variable

1. The Hölder and Minkowski Inequalities for p, q < 1

2. Integrals of Complex-Valued Functions

3. The Expansion of Complex-Valued Functions in Orthogonal Series

XI. Measure in the Plane and in Space

1. Definition and Properties

2. Plane Measure and Linear Measure

XII. Multiple Integrals

1. Definition and Fundamental Properties

2. Multiple Integrals and Iterated Integrals

3. The Double Integral on Unbounded Sets

4. Applications

XIII. The Stieltjes Integral

1. Definition and Existence

2. Integration by Parts and the Limit of Integrals

3. Relation Between the Stieltjes Integral and Lebesgue Integral

Literature

Index

## Description

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.

## Details

- No. of pages:
- 176

- Language:
- English

- Copyright:
- © Pergamon 1961

- Published:
- 1st January 1961

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483280332