## Measure and Integral: An Introduction to Real Analysis by Richard L. Wheeden

this is the book Measure and Integral: An Introduction to Real Analysis in pdf written  by  Richard L. Wheeden , published by Chapman & Hall ,2015  of professors of  science faculties  universities  .

Language of the book: English language

Book Title: Measure and Integral: An Introduction to Real Analysis

Scriptwriter: by Richard L. Wheeden

Year of printing:   Chapman & Hall ,2015

File Format:PDF

Number of chapters:
13 CHAPTER

Number of pages: 534 pages

File Size: 25,10 MB

Contents

Preliminaries

Points and Sets in Rn

Rn as a Metric Space

Open and Closed Sets in Rn, and Special Sets

Compact Sets and the Heine–Borel Theorem

Functions

Continuous Functions and Transformations

The Riemann Integral

Exercises

Functions of Bounded Variation and the Riemann–Stieltjes Integral

Functions of Bounded Variation

Rectifiable Curves

The Riemann–Stieltjes Integral

Exercises

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 Rn

A Nonmeasurable Set

Exercises

Lebesgue Measurable Functions

Elementary Properties of Measurable Functions

Semicontinuous Functions

Properties of Measurable Functions and Theorems of Egorov and Lusin

Convergence in Measure

Exercises

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 Lp Spaces, 0 < p < ∞

Riemann and Lebesgue Integrals

Exercises

Repeated Integration

Fubini’s Theorem

Tonelli’s Theorem

Applications of Fubini’s Theorem

Exercises

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 Rn

Exercises

Lp Classes

Definition of Lp

Hölder’s Inequality and Minkowski’s Inequality

Classes l p

Banach and Metric Space Properties

The Space L2 and Orthogonality

Fourier Series and Parseval’s Formula

Hilbert Spaces

Exercises

Approximations of the Identity and Maximal Functions

Convolutions

Approximations of the Identity

The Hardy–Littlewood Maximal Function

The Marcinkiewicz Integral

Exercises

Abstract Integration

Measurable Functions and Integration

Absolutely Continuous and Singular Set Functions and Measures

The Dual Space of Lp

Relative Differentiation of Measures

Exercises

Outer Measure and Measure

Constructing Measures from Outer Measures

Metric Outer Measures

Lebesgue–Stieltjes Measure

Hausdorff Measure

Carathéodory–Hahn Extension Theorem

Exercises

A Few Facts from Harmonic Analysis

Trigonometric Fourier Series

Convergence of S[f] and SÞ[f]

Divergence of Fourier Series

Summability of Sequences and Series

Summability of S[f] and SÞ[f] by the Method of the Arithmetic Mean

Summability of S[f] by Abel Means

Existence of f Þ

Properties of f Þ for f ∈ Lp, 1 < p < ∞

Application of Conjugate Functions to Partial Sums of S[f]

Exercises

The Fourier Transform

The Fourier Transform on L1

The Fourier Transform on L2

The Hilbert Transform on L2

The Fourier Transform on Lp, 1 < p < 2

Exercises

Fractional Integration

Subrepresentation Formulas and Fractional Integrals

L1, L1 Poincaré Estimates and the Subrepresentation Formula; Hölder Classes

Norm Estimates for Iα

Exponential Integrability of Iαf

Bounded Mean Oscillation

Exercises

Weak Derivatives and Poincaré–Sobolev Estimates

Weak Derivatives

Approximation by Smooth Functions and Sobolev Spaces

Poincaré–Sobolev Estimates

Exercises