This text is designed for graduate-level courses in real analysis.
Real Analysis, 4th Edition, covers the basic material that every graduate student 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 undergraduate mathematics and familiarity with the material covered in an undergraduate course on the fundamental concepts of analysis.
PART I: LEBESGUE INTEGRATION FOR FUNCTIONS OF A SINGLE REAL VARIABLE 1. The Real Numbers: Sets, Sequences and Functions 1.1 The Field, Positivity and Completeness Axioms 1.2 The Natural and Rational Numbers 1.3 Countable and Uncountable Sets 1.4 Open Sets, Closed Sets and Borel Sets of Real Numbers 1.5 Sequences of Real Numbers 1.6 Continuous Real-Valued Functions of a Real Variable 2. Lebesgue Measure 2.1 Introduction 2.2 Lebesgue Outer Measure 2.3 The -algebra of Lebesgue Measurable Sets 2.4 Outer and Inner Approximation of Lebesgue Measurable Sets 2.5 Countable Additivity and Continuity of Lebesgue Measure 2.6 Nonmeasurable Sets 2.7 The Cantor Set and the Cantor-Lebesgue Function 3. Lebesgue Measurable Functions 3.1 Sums, Products and Compositions 3.2 Sequential Pointwise Limits and Simple Approximation 3.3 Littlewood's Three Principles, Egoroff's Theorem and Lusin's Theorem 4. Lebesgue Integration 4.1 The Riemann Integral 4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure 4.3 The Lebesgue Integral of a Measurable Nonnegative Function 4.4 The General Lebesgue Integral 4.5 Countable Additivity and Continuity of Integraion 4.6 Uniform Integrability: The Vitali Convergence Theorem 5. Lebesgue Integration: Further Topics 5.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem 5.2 Convergence in measure 5.3 Characterizations of Riemann and Lebesgue Integrability 6. Differentiation and Integration 6.1 Continuity of Monotone Functions 6.2 Differentiability of Monotone Functions: Lebesgue's Theorem 6.3 Functions of Bounded Variation: Jordan's Theorem 6.4 Absolutely Continuous Functions 6.5 Integrating Derivatives: Differentiating Indefinite Integrals 6.6 Convex Functions 7. The L Spaces: Completeness and Approximation 7.1 Normed Linear Spaces 7.2 The Inequalities of Young, Hoelder and Minkowski 7.3 L is Complete: The Riesz-Fischer Theorem 7.4 Approximation and Separability 8. The L Spaces: Duality and Weak Convergence 8.1 The Dual Space of L 8.2 Weak Sequential Convergence in L 8.3 Weak Sequential Compactness 8.4 The Minimization of Convex Functionals PART II: ABSTRACT SPACES: METRIC, TOPOLOGICAL, AND HILBERT 9. Metric Spaces: General Properties 9.1 Examples of Metric Spaces 9.2 Open Sets, Closed Sets and Convergent Sequences 9.3 Continuous Mappings Between Metric Spaces 9.4 Complete Metric Spaces 9.5 Compact Metric Spaces 9.6 Separable Metric Spaces 10. Metric Spaces: Three Fundamental Theorems 10.1 The Arzela-Ascoli Theorem 10.2 The Baire Category Theorem 10.3 The Banach Contraction Principle 11. Topological Spaces: General Properties 11.1 Open Sets, Closed Sets, Bases and Subbases 11.2 The Separation Properties 11.3 Countability and Separability 11.4 Continuous Mappings Between Topological Spaces 11.5 Compact Topological Spaces 11.6 Connected Topological Spaces 12. Topological Spaces: Three Fundamental Theorems 12.1 Urysohn's Lemma and the Tietze Extension Theorem 12.2 The Tychonoff Product Theorem 12.3 The Stone-Weierstrass Theorem 13. Continuous Linear Operators Between Banach Spaces 13.1 Normed Linear Spaces 13.2 Linear Operators 13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces 13.4 The Open Mapping and Closed Graph Theorems 13.5 The Uniform Boundedness Principle 14. Duality for Normed Linear Spaces 14.1 Linear Functionals, Bounded Linear Functionals and Weak Topologies 14.2 The Hahn-Banach Theorem 14.3 Reflexive Banach Spaces and Weak Sequential Convergence 14.4 Locally Convex Topological Vector Spaces 14.5 The Separation of Convex Sets and Mazur's Theorem 14.6 The Krein-Milman Theorem 15. Compactness Regained: The Weak Topology 15.1 Alaoglu's Extension of Helley's Theorem 15.2 Reflexivity and Weak Compactness: Kakutani's Theorem 15.3 Compactness and Weak Sequential Compactness: The Eberlein-Smulian Theorem 15.4 Metrizability of Weak Topologies 16. Continuous Linear Operators on Hilbert Spaces 16.1 The Inner Product and Orthogonality 16.2 The Dual Space and Weak Sequential Convergence 16.3 Bessel's Inequality and Orthonormal Bases 16.4 Adjoints and Symmetry for Linear Operators 16.5 Compact Operators 16.6 The Hilbert Schmidt Theorem 16.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators PART III: MEASURE AND INTEGRATION: GENERAL THEORY 17. General Measure Spaces: Their Properties and Construction 17.1 Measures and Measurable Sets 17.2 Signed Measures: The Hahn and Jordan Decompositions 17.3 The Caratheodory Measure Induced by an Outer Measure 17.4 The Construction of Outer Measures 17.5 The Caratheodory-Hahn Theorem: The Extension of a Premeasure to a Measure 18. Integration Over General Measure Spaces 18.1 Measurable Functions 18.2 Integration of Nonnegative Measurable Functions 18.3 Integration of General Measurable Functions 18.4 The Radon-Nikodym Theorem 18.5 The Saks Metric Space: The Vitali-Hahn-Saks Theorem 19. General L Spaces: Completeness, Duality and Weak Convergence 19.1 The Completeness of L ( , ), 1 19.2 The Riesz Representation theorem for the Dual of L ( , ), 1 19.3 The Kantorovitch Representation Theorem for the Dual of L ( , ) 19.4 Weak Sequential Convergence in L (X, ), 1 < < 1 19.5 Weak Sequential Compactness in L1 (X, ): The Dunford-Pettis Theorem 20. The Construction of Particular Measures 20.1 Product Measures: The Theorems of Fubini and Tonelli 20.2 Lebesgue Measure on Euclidean Space Rn 20.3 Cumulative Distribution Functions and Borel Measures on R 20.4 Caratheodory Outer Measures and hausdorff Measures on a Metric Space 21. Measure and Topology 21.1 Locally Compact Topological Spaces 21.2 Separating Sets and Extending Functions 21.3 The Construction of Radon Measures 21.4 The Representation of Positive Linear Functionals on Cc (X): The Riesz-Markov Theorem 21.5 The Riesz Representation Theorem for the Dual of C(X) 21.6 Regularity Properties of Baire Measures 22. Invariant Measures 22.1 Topological Groups: The General Linear Group 22.2 Fixed Points of Representations: Kakutani's Theorem 22.3 Invariant Borel Measures on Compact Groups: von Neumann's Theorem 22.4 Measure Preserving Transformations and Ergodicity: the Bogoliubov-Krilov Theorem