Integral, Measure and Derivative: A Unified Approach (Dover Books on Mathematics) by G. E. Shilov (PDF)

Description

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.

User’s Reviews

Editorial Reviews: 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.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is very much a graduate level text. It says on the back “… this book can be understood by any reader with a good background in advanced calculus and with sufficient ‘mathematical maturity’.” Do not let this deceive you, the layman will not be able to adequately digest this book. Also, it is a Russian textbook, and anyone well versed in mathematics will know that Russian textbooks are always the most unforgiving. If this is your first time dealing with measure theory and Lebesgue integration, I would be prepared to spend hours upon hours on these pages. I am giving 4 stars because I believe that this book is extremely dense with information, but is also fairly dry. It throws you right into the material from the beginning and does not slow down to help the reader along, that is purely your own responsibility. So, if you are used to having your hand held then I would look at more easygoing material on this subject before working on this text. The approach (Daniell scheme) is entirely unique, though, and is the biggest advantage of this text.

⭐The theory of measure and integral or “Lebesque measure theory” is a standard beginning graduate course in most decent mathematics schools, and it is an upper division course in very good undergraduate programs. Royden’s text was very standard for a long time, although I think it is a very bad textbook. Ruden is better, but the theory still seems ponderous to me. Of the standard textbooks, I really think that Malliavin is by far the best. His constructions are complete and correct, and he makes the serious connections to the rest of analysis that are required. The volume by Stein is also very good.This book gives what is to me a much quicker, much cleaner, and much more insightful construction of modern measure theory. The development extends the Riemann integral (freshman calculus) via the Daniell construction, and it produces the full integration theory and measure along with it in quite a direct and intuitive way. It is really “the right” way to do it, and I’ve never seen it anywhere else. I think that an undergrad course would need to use Stein’s text. A graduate course should, I think, use Malliavin. In either case, I think this is an indispensable co-text. It will allow anyone to get a full understanding of the whole area very quickly, very cleanly, and without a lot of trouble.

⭐Make sure you read the comment “Be wary”, it is a graduate level text. But it will bring you straight to the subject in a clear and/but concise way.

⭐It is a very good book about measure theory.

⭐The traditional approach to define Lebesgue integration is long and boring, in my opinion. This book is rather unique in its approach to introduce the Lebesgue integration using the Daniell integral, see […] . This makes it short, concise, and very easy. If you want to learn and understand both Riemann and Lebesgue integrals in a week, just read this book. One more week reading it, and you now know the Riemann-Stieltjes and Lebesgue-Stieltjes integration as well, no extra charge!Hopefully, Dover prints more copies of it, as it gets hard to find even used.

⭐Contenuto veramente molto buono … L’ unica pecca è la scarsa qualità della stampa .. Ed un appunto alla dover … Curate la qualità della stampa e la rilegatura .. Un tempo facevate ottimi libri ..

⭐

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