Real analysis: an introduction to the theory of real functions and integration

Abstract

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