This book giving an exposition of the foundations of modern measure theory offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course, and, finally, more specialized topics partly covered by more than 850 exercises with detailed hints and references. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided.

Table of Contents

Chapter 1. Constructions and extensions of measures

Chapter 2. The Lebesgue integral

Chapter 3. Operations on measures and functions

Chapter 4. The spaces LA p and spaces of measures

Chapter 5. Connections between the integral and derivative

Review

From the reviews:

"The main thrust of the very detailed account of this subject by Bogachev in two volumes making up approximately 1100 pages, with 2038 references listed, is a scholarly rendering of its many sided view, some highlights of which will be commented on here. … The treatment is reader friendly and I would recommend that each graduate real analysis student own both volumes … and they make a good reference set to keep on ones shelf." (Malempati M. Rao, Zentralblatt MATH, Vol. 1120 (22), 2007)

"Volume one contains the modern foundations of measure and integration theory … . I should mention that the problems are accessible to students―some of the exercises are especially recommended for students … . The monograph excels by its clear, scholarly style and the wealth of … historical comments and references. … For any library and researchers in mathematical analysis or probability theory these … are a must-have―for the mathematical literature this is a wonderful addition." (René L. Schilling, Mathematical Reviews, Issue 2008 g)

"This is a remarkably comprehensive treatise on modern, as well as classical, measure theory and integration. … This is an excellent and impressive monograph, which I can strongly recommended to researchers in analysis and probability, to university teachers as well as to students. I am convinced that this … volume treatise cannot be missing from university libraries and the shelves of mathematicians interested in measure and integration." (EMS Newsletter, December, 2008)

About the Author

Vladimir Bogachev was born in Moscow in 1961. He got the PhD at Moscow State University in 1986 and he got the degree of Doctor of Sciences in 1990. Since 1986 Vladimir Bogachev has worked at the Department of Mechanics and Mathematics of Moscow State University. The main fields of his research are measure theory, nonlinear functional analysis, probability theory, and stochastic analysis. He is a well-nown expert in measure theory, probability theory, and the Malliavin calculus, and the author of more than 100 scientific publications. His monograph ``Gaussian Measures’’ (AMS, 1998) has become a widely used source. Vladimir Bogachev hs been an invited speaker and a lecturer at many international conferences and several dozen universities and mathematical institutes over the world.

Scientific awards: a medal of the Academy of Sciences of the USSR and the Award of the Japan Society of Promotion of Science.