Real analysis (International Edition) (4^th Ed.)

Real Analysis, Fourth 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. Patrick Fitzpatrick of the University of Maryland—College Park spearheaded this revision of Halsey Royden's classic text.

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 σ-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^Ρ Spaces: Completeness and Approximation

7.1 Normed Linear Spaces

7.2 The Inequalities of Young, Hölder and Minkowski

7.3 L^Ρ is Complete: The Riesz-Fischer Theorem

7.4 Approximation and Separability

8. The L^Ρ Spaces: Duality and Weak Convergence

8.1 The Dual Space of L^Ρ

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