Riemann integral

Abstract: In this paper a pair of Gauss-Chebyshev integration formulas for singular integrals are developed. Using these formulas a simple numerical method for solving a system of singular integral equations is described. To demonstrate the effectiveness of the method, a numerical example is given. In order to have a basis of comparison, the example problem is solved also by using an alternate method.

Abstract: Preface. 1. Integration of Monotone Functions on Intervals. 2. Set Functions and Caratheodory Measurability. 3. Construction of Measures using Topology. 4. Distribution Functions, Measurability and Comonotonicity of Functions. 5. The Asymmetric Integral. 6. The Subadditivity Theorem. 7. The Symmetric Integral. 8. Sequences of Functions and Convergence Theorems. 9. Nullfunctions and the Lebesgue Spaces Lp. 10. Families of Measures and their Envelopes. 11. Densities and the Radon-Nikodym Theorem. 12. Products. 13. Representing Functionals as Integrals. References. Index.

Abstract: Preface to the Second Edition Preface to the First Edition Authors 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 Further Results about Riemann-Stieltjes Integrals 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 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 Holder'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 Additive Set Functions and Measures 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 Caratheodory-Hahn Extension Theorem Exercises A Few Facts from Harmonic Analysis Trigonometric Fourier Series Theorems about Fourier Coefficients Convergence of S[f] and SP[f] Divergence of Fourier Series Summability of Sequences and Series Summability of S[f] and SP[f] by the Method of the Arithmetic Mean Summability of S[f] by Abel Means Existence of f P Properties of f P for f Lp, 1 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 2 Exercises Fractional Integration Subrepresentation Formulas and Fractional Integrals L1, L1 Poincare Estimates and the Subrepresentation Formula Holder Classes Norm Estimates for Ialpha Exponential Integrability of Ialphaf Bounded Mean Oscillation Exercises Weak Derivatives and Poincare-Sobolev Estimates Weak Derivatives Approximation by Smooth Functions and Sobolev Spaces Poincare-Sobolev Estimates Exercises Notations Index