Integral, Measure and Derivative: A Unified Approach

by G. E. Shilov, B. L. Gurevich

Overview

This graduate-level textbook and monograph defines the functions of a real variable through consistent use of the Daniell scheme, offering a rare and useful alternative to customary approaches. The treatment can be understood by any reader with a solid background in advanced calculus, and it features many problems with hints and answers. "The exposition is fresh and sophisticated," declared Sci-Tech Book News, "and will engage the interest of accomplished mathematicians."
Part one is devoted to the integral, moving from the Reimann integral and step functions to a general theory, and obtaining the "classical" Lebesgue integral in n space. Part two constructs the Lebesgue-Stieltjes integral through the Daniell scheme using the Reimann-Stieltjes integral as the elementary integral. Part three develops theory of measure with the general Daniell scheme, and the final part is devoted to the theory of the derivative.

Product Details

ISBN-13:	9780486165615
Publisher:	Dover Publications
Publication date:	05/13/2013
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	256
File size:	13 MB
Note:	This product may take a few minutes to download.

About the Author

Georgi Evgen'evich Shilov (1917–75) was a Soviet mathematician and an expert in the field of functional analysis who contributed to the theory of normed rings and generalized functions. He is the author of five Dover books.
B. L. Gurevich was a Soviet mathematician.

Table of Contents

IntroductionPart 1 The Integral1. The Riemann Integral and Step Functions2. General Theory of the Integral3. The Lebesgue Integral in n-SpacePart 2 The Stieltjes Integral4. The Riemann-Stieltjes Integral5. The Lebesgue-Stieltjes Integral6. Measurable Sets and General Measure Theory7. Construtive Measure Theory8. Axiomatic Measure Theory9. Measure and Set Functions10. The Derivative of a Set FunctionBibliographyIndex