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.
Table of Contents
PART I. AN INTRODUCTION TO GENERAL TOPOLOGY
SET-THEORETIC AND ALGEBRAIC PRELIMINARIES
Sets and Basic Notation
Set Operations under Maps
Relations and Well-Ordering Principle
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
Linear and Normed Linear Spaces
ELEMENTS OF POINT SET TOPOLOGY
Bases and Subbases for Topological Spaces
Convergence of Sequences in Topological Spaces and
Continuity in Topological Spaces
Notes on Subspaces and Compactnes
Function Spaces and Ascoli's Theorem
Stone-Weierstrass Approximation Theorem
Filter and Net Convergence
Functions on Locally Compact Spaces
PART II. BASICS OF MEASURE AND INTEGRATION
MEASURABLE SPACES AND MEASURABLE FUNCTIONS
Systems of Sets
Extension of Set Functions to a Measure
Lebesgue and Lebesgue-Stieltjes Measures
Extended Real-Valued Measurable 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.
Product Measures of Finitely Many Measurable Spaces and
Applications of Fubini's Theorem
CALCULUS IN EUCLIDEAN SPACES
Change of Variables
PART III. FURTHER TOPICS IN INTEGRATION
ANALYSIS IN ABSTRACT SPACES
Signed and Complex Measures
Modes of Convergence
Radon Measures on Locally Compact Hausdorff Spaces
CALCULUS ON THE REAL LINE
Functions of Bounded Variation
Absolute Continuous Functions
"…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."
"…offers the perfect vehicle for building the foundation needed for more advanced studies."
--Mathematical Reviews, Issue 2001h