Measure and Integral (Chapman & Hall/CRC Pure and Applied Mathematics)

Measure and Integral (Chapman & Hall/CRC Pure and Applied Mathematics)

<P>Now considered a classic text on the topic,<B> Measure and Integral: An Introduction to Real Analysis</B> 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.</P> <P>Published nearly forty years after the first edition, this long-awaited <B>Second Edition </B>also:</P> <UL> <LI>Studies the Fourier transform of functions in the spaces <I>L<SUP>1</I></SUP>, <I>L<SUP>2</I></SUP>, and <I>L<SUP>p</I></SUP>, 1 < <I>p</I> < 2</LI> <LI>Shows the Hilbert transform to be a bounded operator on <I>L<SUP>2</I></SUP>, as an application of the <I>L<SUP>2</I></SUP> theory of the Fourier transform in the one-dimensional case</LI> <LI>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</LI> <LI>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</LI> <LI>Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient</LI> <LI>Applies the norm estimates derived for fractional integral operators to obtain local and global first-order Poincaré–Sobolev inequalities, including endpoint cases</LI> <LI>Proves the existence of a tangent plane to the graph of a Lipschitz function of several variables</LI> <LI>Includes many new exercises not present in the first edition</LI></UL> <P>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.</P>