Royden H.I., Fitzpatrick P. Real Analysis

4th Edition. — Peason Education Asia Lim., 2010. — 516 p. — ISBN: 013143747X.Real Analysis, Fourth Edition, covers the basic material that every reader should know in the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. This text assumes a general background in mathematics and familiarity with the fundamental concepts of analysis.
Classical theory of functions, including the classical Banach spaces; General topology and the theory of general Banach spaces; Abstract treatment of measure and integration.Lebesgue Integration for Functions of a Single Real Variable
The Real Numbers: Sets, Sequences, and Functions
Lebesgue Measure
Lebesgue Measurable Functions
Lebesgue Integration
Lebesgue Integration: Further Topics
Differentiation and Integration
The I^p Spaces: Completeness and Approximation
The I^p Spaces: Duality and Weak Convergence
Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces
Metric Spaces: General Properties
Metric Spaces: Three Fundamental Theorems
Topological Spaces: General Properties
Topological Spaces: Three Fundamental Theorems
Continuous Linear Operators Between Banach Spaces
Duality for Normed Linear Spaces
Compactness Regained: The Weak Topology
Continuous Linear Operators on Hilbert Spaces
Measure and Integration: General Theory
General Measure Spaces: Their Properties and Construction
Integration Over General Measure Spaces
General LP Spaces: Completeness, Duality, and Weak Convergence
The Construction of Particular Measures
Measure and Topology
Invariant Measures