Real Analysis

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

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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 more

Product details

  • Hardback | 584 pages
  • 159.8 x 241 x 31mm | 934.42g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • New.
  • 1584880732
  • 9781584880738

Review quote

"bound to become a classic for studentsbecause 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 2001hshow more

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 BIBLIOGRAPHYshow more