Real Analysis: An Introduction to the Theory of Real Functions and Integration, 1st Edition (Hardback) book cover

Real Analysis

An Introduction to the Theory of Real Functions and Integration, 1st Edition

By Jewgeni H. Dshalalow

Chapman and Hall/CRC

584 pages

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pub: 2000-09-28
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Description

Designed for use in a two-semester course on abstract analysis, REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration illuminates the principle topics that constitute real analysis. Self-contained, with coverage of topology, measure theory, and integration, it offers a thorough elaboration of major theorems, notions, and constructions needed not only by mathematics students but also by students of statistics and probability, operations research, physics, and engineering.

Structured logically and flexibly through the author's many years of teaching experience, the material is presented in three main sections:

Part 1, chapters 1through 3, covers the preliminaries of set theory and the fundamentals of metric spaces and topology. This section can also serves as a text for first courses in topology.

Part II, chapter 4 through 7, details the basics of measure and integration and stands independently for use in a separate measure theory course.

Part III addresses more advanced topics, including elaborated and abstract versions of measure and integration along with their applications to functional analysis, probability theory, and conventional analysis on the real line.

Analysis lies at the core of all mathematical disciplines, and as such, students need and deserve a careful, rigorous presentation of the material. REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration offers the perfect vehicle for building the foundation students need for more advanced studies.

Reviews

"…bound to become a classic for students…because it is pleasant to use, and because all classical results on measure and integration are completely covered."

Gustave Choquet

"…offers the perfect vehicle for building the foundation needed for more advanced studies."

--Mathematical Reviews, Issue 2001h

Table of Contents

PART I. AN INTRODUCTION TO GENERAL TOPOLOGY

SET-THEORETIC AND ALGEBRAIC PRELIMINARIES

Sets and Basic Notation

Functions

Set Operations under Maps

Relations and Well-Ordering Principle

Cartesian Product

Cardinality

Basic Algebraic Structures

ANALYSIS OF METRIC SPACES

Definitions and Notations

The Structure of Metric Spaces5

Convergence in Metric Spaces

Continuous Mappings in Metric Spaces

Complete Metric Spaces

Compactness

Linear and Normed Linear Spaces

ELEMENTS OF POINT SET TOPOLOGY

Topological Spaces

Bases and Subbases for Topological Spaces

Convergence of Sequences in Topological Spaces and

Countability

Continuity in Topological Spaces

Product Topology

Notes on Subspaces and Compactnes

Function Spaces and Ascoli's Theorem

Stone-Weierstrass Approximation Theorem

Filter and Net Convergence

Separation

Functions on Locally Compact Spaces

PART II. BASICS OF MEASURE AND INTEGRATION

MEASURABLE SPACES AND MEASURABLE FUNCTIONS

Systems of Sets

System's Generators

Measurable Functions

MEASURES

Set Functions

Extension of Set Functions to a Measure

Lebesgue and Lebesgue-Stieltjes Measures

Image Measures

Extended Real-Valued Measurable Functions

Simple Functions

ELEMENTS OF INTEGRATION

Integration on C -1(W,S)

Main Convergence Theorems

Lebesgue and Riemann Integrals on R

Integration with Respect to Image Measures

Measures Generated by Integrals. Absolute Continuity.

Orthogonality

Product Measures of Finitely Many Measurable Spaces and

Fubini's Theorem

Applications of Fubini's Theorem

CALCULUS IN EUCLIDEAN SPACES

Differentiation

Change of Variables

PART III. FURTHER TOPICS IN INTEGRATION

ANALYSIS IN ABSTRACT SPACES

Signed and Complex Measures

Absolute Continuity

Singularity

Lp Spaces

Modes of Convergence

Uniform Integrability

Radon Measures on Locally Compact Hausdorff Spaces

Measure Derivatives

CALCULUS ON THE REAL LINE

Monotone Functions

Functions of Bounded Variation

Absolute Continuous Functions

Singular Functions

BIBLIOGRAPHY

About the Series

Studies in Advanced Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT037000
MATHEMATICS / Functional Analysis
SCI040000
SCIENCE / Mathematical Physics
TEC029000
TECHNOLOGY & ENGINEERING / Operations Research