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Full Description
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 < p < 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 Hölder 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 Poincaré-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.
Contents
Preliminaries. Functions of Bounded Variation and the Riemann-Stieltjes Integral. Lebesgue Measure and Outer Measure. Lebesgue Measurable Functions. The Lebesgue Integral. Repeated Integration. Differentiation. Lp Classes. Approximations of the Identity and Maximal Functions. Abstract Integration. Outer Measure and Measure. A Few Facts from Harmonic Analysis. The Fourier Transform. Fractional Integration. Weak Derivatives and Poincaré-Sobolev Estimates.
