Preface to the Second Edition Preface to the First Edition Authors
1.**Preliminaries**
Points and Sets in R^{n}R^{n} as a Metric Space Open and Closed Sets in R^{n}, and Special Sets Compact Sets and the Heine–Borel Theorem Functions Continuous Functions and Transformations The Riemann Integral Exercises
**2.Functions of Bounded Variation and the Riemann–Stieltjes Integral**
Functions of Bounded Variation Rectifiable Curves The Riemann–Stieltjes Integral Further Results about Riemann–Stieltjes Integrals Exercises
**3.Lebesgue Measure and Outer Measure**
Lebesgue Outer Measure and the Cantor Set Lebesgue Measurable Sets Two Properties of Lebesgue Measure Characterizations of Measurability Lipschitz Transformations of R^{n}A Nonmeasurable Set Exercises
**4.Lebesgue Measurable Functions**
Elementary Properties of Measurable Functions Semicontinuous Functions Properties of Measurable Functions and Theorems of Egorov and Lusin Convergence in Measure Exercises
**5.The Lebesgue Integral**
Definition of the Integral of a Nonnegative Function Properties of the Integral The Integral of an Arbitrary Measurable* f*
Relation between Riemann–Stieltjes and Lebesgue Integrals, and the *L*^{p} Spaces, 0 < *p* < ¡Ä Riemann and Lebesgue Integrals Exercises
**6.Repeated Integration**
Fubini¡¯s Theorem Tonelli¡¯s Theorem Applications of Fubini¡¯s Theorem Exercises
**7.Differentiation**
The Indefinite Integral Lebesgue¡¯s Differentiation Theorem Vitali Covering Lemma Differentiation of Monotone Functions Absolutely Continuous and Singular Functions Convex Functions The Differential in R^{n}Exercises
*8.L*^{p} Classes
Definition of *L*^{p}Hölder¡¯s Inequality and Minkowski¡¯s Inequality Classes* l *^{p}Banach and Metric Space Properties The Space *L*^{2} and Orthogonality Fourier Series and Parseval¡¯s Formula Hilbert Spaces Exercises
**9.Approximations of the Identity and Maximal Functions**
Convolutions Approximations of the Identity The Hardy–Littlewood Maximal Function The Marcinkiewicz Integral Exercises
**10.Abstract Integration**
Additive Set Functions and Measures Measurable Functions and Integration Absolutely Continuous and Singular Set Functions and Measures The Dual Space of *L*^{p}Relative Differentiation of Measures Exercises
**11.Outer Measure and Measure**
Constructing Measures from Outer Measures Metric Outer Measure Lebesgue–Stieltjes Measure Hausdorff Measure Carathéodory–Hahn Extension Theorem Exercises
**12.A Few Facts from Harmonic Analysis**
Trigonometric Fourier Series Theorems about Fourier Coefficients Convergence of *S*[*f*] and *S¨*[*f*] Divergence of Fourier Series Summability of Sequences and Series Summability of *S*[*f*] and *S¨*[*f*] by the Method of the Arithmetic Mean Summability of *S*[*f*] by Abel Means Existence of *f ¨*
Properties of* f ¨* for *f* ¡ô *L*^{p}, 1 < *p* < ¡Ä Application of Conjugate Functions to Partial Sums of *S*[*f*] Exercises
**13.The Fourier Transform**
The Fourier Transform on *L*^{1}The Fourier Transform on *L*^{2}The Hilbert Transform on *L*^{2}The Fourier Transform on *L*^{p}, 1 < *p* < 2 Exercises
**14.Fractional Integration**
Subrepresentation Formulas and Fractional Integrals
*L*^{1}, L^{1} Poincaré Estimates and the Subrepresentation Formula; Hölder Classes Norm Estimates for *I*_{¥á}Exponential Integrability of *I*_{¥á}f
Bounded Mean Oscillation Exercises
**15.Weak Derivatives and Poincaré–Sobolev Estimates**
Weak Derivatives Approximation by Smooth Functions and Sobolev Spaces Poincaré–Sobolev Estimates Exercises
Notations
Index |