This volume develops the classical theory of the Lebesgue integral and some of its applications. The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given. Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis--harmonic analysis--are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function. Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas. 

Preliminaries Points and Sets in Rn Rn as a Metric Space Open and Closed Sets in Rn: Special Sets Compact Sets; The Heine-Borel Theorem Functions Continuous Functions and Transformations The Riemann Integral Exercises Function of Bounded Variation; The Riemann-Stieltjes Integral Functions of Bounded Variation Rectifiable Curves The Reiman-Stieltjes Integral Further Results About the Reimann-Stieltjes Integrals Exercises Lebesgue Measure and Outer Measure Lebesgue Outer Measures; 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; Egorov's Theorem and Lusin's Theorem 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 A Relation Between Riemann-Stieltjes and Lebesgue Integrals; the LP Spaces, 0

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