| I was turned on to this book by a friend of mine who is an expert in geometric measure theory. He recommended the book as a very nice exposition of some of the material found in Federer's "Geometric Measure Theory" as well as other material. I found the book to be beautifully designed to help the reader learn its contents. There was enough between the lines so that one needed to WORK through the book, but in contrast to parts of Federer's book, enough detail so that reasonably fast progress could be made. Unfortunately, I was interupted in my race through the book and so I have yet to work through the latter part of the book. But given the large part I did cover and my experience doing that, I am certain to finish the monograph, most likely when I start using functions of bounded variation with any frequency. There are no explicit exercises. But as already alluded to above, there are implicit exercises that are encountered in working through the book. I found that the lack of separate exercises is actually not bad at all since the implicit exercises encountered are automatically motivated by their necessity for the understanding of the text - and are therefore relevant! A prerequisite for the book is a course in analysis that includes measure theory and integration as well as an exposure to elementary functional analysis. The functional analysis is not actually necessary, but the added maturity that such an exposure would impart would be useful. Very briefly, the contents via the 6 chapter titles are 1) General Measure Theory, 2) Hausdorff Measure, 3) Area and Coarea Formulas, 4) Sobolev Functions, 5) BV Functions and Sets of Finite Perimeter, and 6) Differentiability and Approximation by C^1 Functions. I found the contents very interesting ... quoting the authors "... we packed into these notes all sorts of interesting topics that working mathematical analysts need to know, but are mostly not taught." And indeed this was the case in my experience ... both the "interesting" part and the "not taught" part. I am disappointed in the price, but if any book is worth it, this one certainly is. |
| This book provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space and emphasizes the roles of Hausdorff measure and the capacity in characterizing the find properties of sets and functions. Topics covered include a quick review of abstract measure theory, theorems and differentiation in Mn, lower Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions and functions of bounded variation. The text provides complete proofs of many key results omitted from other books, including Besicovitch's Covering Theorem, Rademacher's Theorem (on the differentiability a.e. of Lipschitz functions), the Area and Coarea Formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Alexandro's Theorem (on the twice differentiability a.e. of convex functions).Topics are carefully selected and the proofs succinct, but complete, which makes this book ideal reading for applied mathematicians and graduate students in applied mathematics. |
