Measure and integral : an introduction to real analysis Richard L. Wheeden.

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