Real Analysis An Introduction to the Theory of Real Functions and Integration

REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration is intended to be used in a two-semester course on Real Analysis/Real Variables at the graduate level. It is self-contained, with coverage included on topology, measure theory and integration. It includes many applications from the fields of science and engineering, thus serving a broad audience. The topics addressed include Ascoli's theorem, Stone-Weierstrass approximation, and applications of Fubini's theorem.

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.

Set-Theoretic and Algebraic Preliminaries. Analysis of Metric Spaces. Elements of Point Set Topology. Measurable Spaces and Measurable Functions. Measures. Elements of Integration. Calculus in Euclidean Spaces. Further Topics in Integration. Calculus on the Real Line.