Measure and Integral: An Introduction to Real Analysis

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Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less geometric content. Published nearly forty years after the first edition, this long-awaited Second Edition also: * Studies the Fourier transform of functions in the spaces L1, L2, and Lp, 1 2 * Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case * Covers fractional integration and some topics related to mean oscillation properties of functions, such as the classes of Holder continuous functions and the space of functions of bounded mean oscillation * Derives a subrepresentation formula, which in higher dimensions plays a role roughly similar to the one played by the fundamental theorem of calculus in one dimension * Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient * Applies the norm estimates derived for fractional integral operators to obtain local and global first-order Poincare-Sobolev inequalities, including endpoint cases * Proves the existence of a tangent plane to the graph of a Lipschitz function of several variables * Includes many new exercises not present in the first edition This widely used and highly respected text for upper-division undergraduate and first-year graduate students of mathematics, statistics, probability, or engineering is revised for a new generation of students and instructors. The book also serves as a handy reference for professional mathematicians.