Lebesgue Integration on Euclidean Space, Revised Edition (Jones and Bartlett Books in Mathematics) by Frank Jones (PDF)

Description

Lebesgue Integration on Euclidean Space contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. Throughout the text, many exercises are incorporated, enabling students to apply new ideas immediately. Jones strives to present a slow introduction to Lebesgue integration by dealing with n-dimensional spaces from the outset. In addition, the text provides students a thorough treatment of Fourier analysis, while holistically preparing students to become workers in real analysis.

User’s Reviews

Editorial Reviews: About the Author Frank Jones, Rice University Frank Jones received his bachelor’s and Ph.D. degrees from Rice University. His major research interest include real analysis and partial differential equations. In addition, he has been awarded several distinguished teaching awards throughout his career at Rice. Frank’s current intereest are partial differential equations and real analysis.

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

⭐[This review has been completely rewritten on October 24, and the ratings have been incremented from 3 stars to 4]The following review is based on my impression from the first five chapters of the book.This is a great book for those who are already acquainted with the material. However it is not adequate for self-study for the following three reasons:1. No official solutions manual is available, either in print or online. This point is particularly painful as the exercises are intimately integrated in the main text, in the sense that results stated in exercises are occasionally referenced in proofs.2. There is no indication as to the interdependency between the chapters.3. No official errata sheet is available online.

⭐If you want to see measure theory and Lebesgue integration developed in their original real analysis framework look no further. I admit he uses the artifice of special rectangles at first but he generalizes these to the familiar intervals (even uses rotation matrices) and in the end you get the Lebesgue theory from piecewise comprehensible components. The first chapter is a review of the needed real analysis concepts and theorems. There’s a heavy use of set theory and sequences in this chapter. No surprise as set theory and orderings are key to the development. In the proof of the Heine-Borel Theorem he makes use of what he calls the completeness of the real number field as exemplified in the fact (theorem) that a bounded increasing real sequence converges to a limit. Completeness is generally the least upper bound property which is key to proving this sequence theorem (found in chapter 3 of Rudin’s Principles). Within the first few pages he gave an exercise on the lim sup and lim inf of a sequence of sets and this actually involves an ordering by inclusion (a set is viewed as the greater if it contains the other). For example if you take the intersection of a few arbitrary sets and compare it to the intersection with one of those sets left out, this second intersection is the greater. You’ll use these ideas in the exercise along with notions of union and complement. Don’t fret if you can’t do all the exercises-only a few are used in the text development and if necessary can be found online. There was an exercise on lim sup of a real sequence which I had to look up because I learned this was the supremum of its subsequential limits and had never known of an actual construction. This is on p. 60 of chapter 2. Actually it forms a bounded decreasing sequence and so involves the infimum-that’s the part that’s not mentioned in the exercise. This can be found on p. 14 of Rudin’s

⭐. Which book I recommend for supplemental or subsequent reading (Just came out that way!).The first six chapters construct measure theory with the seventh chapter building the integral with the simple functions. From here you can continue onto Fubini’s Theorem which is multiple integration or jump to Rudin whose axiomatic treatment will now seem natural as you’ve already seen inner and outer measures, set algebras, etc. in their real concrete settings. In Rudin you’ll get some things left out of Jones’ fine book like Radon-Nikodym. Of course you could just continue through to the excellent first class treatment of Fourier analysis and differentiation. Or even quit after chapter 7 if you just wanted the basic ideas involved. Bet you’ll continue or go on to Rudin or maybe Halmos!

⭐very very very bad printing, and the problem sections are printed as dark color where I can not read very clearly

⭐One of the problems with modern mathematics is its obsession with rigor which has been attended, over the last few decades, by a mushrooming of symbols and jargon. Much of it is not clearly related to the ideas they serve to label, as evidenced by such terms as the topological use of “filter” whose etymology is obscure (ascribed by some to H. Cartan). Moreover, the particular subject of Lebesgue integration and its generalizations is made even more confusing by a wide variety of approaches depending on an author’s penchants–many of whom are enamored with a purely axiomatic approach and who make little or no appeal to intuition or–God forbid!–pictures. The author of the present work is obviously someone who has actually taught mathematics and taught it lovingly. This book is an excellent read with lots of interesting topics well explained from a student’s point of view. There seems to be a nice ramping from the truly elementary to the sophisticated, which means the book will interest experienced mathematicians, scientists and engineers. There are lots of “doable” problems that the reader can solve along the way. For the experienced mathematician these little problems help alot as a refresher (Oh!, now I remember, that’s how you do it.). I like the emphasis on Euclidean space. Somehow, I always feel more comfortable there! It gives me things I can actually construct and doodle on paper. And, it allows the author to use a few figures in a meaningful way. Which is another of the book’s strong points and if I could recommend a future improvement, it would be to bring on more of those pictures! Tristram Needham has done a nice job along these lines with his book “Visual Complex Analysis.” (I ordered several copies as Christmas gifts–just kidding!). Anyone who has taught mathematics and genuinely wished to be understood by his students has, at various times, drawn them pictures. Inside the cover sheets are lists of integration formulae, a fourier transform table, and a table of “assorted facts” on things like the Gamma function; which show that this is not only a book on Lebesgue integration but a calculus book with the Lebesgue integral occupying center stage. Everyone who has been enamored by the notion of the integral–as I was as a freshman calculus student and have been ever since–will want to have this book on their shelf.

⭐This is no doubt a great book. But the paper quality is really poor.On every printed page, you can see through the other side.At this price, a better paper quality is expected.

⭐* IntroductionThe first thing I need to say I have bought this book from Amazon a long while ago. I paid a lot less than the cash here. Check the lower prices available here.* PhysicalThe book is on the larger size of good quality paper. The font size is helpful for those using specs. You require some in-depth ability to read Greek lettering and mathematical symbols and operations they represent.* Target audienceIts pitched at final year Hons mathematics topic. A math professor I used to know who read this volume said it’s “Easy…” It can be a topic that the Open University support.* Areas coveredThis book covers the real and sets analysis underpinning Calculus. The skill is to see this in two layers. To see this more advanced proving of calculus in how the limit concept is explored and makes you really think ‘Wow!’ as its so beautiful. Basically, its when the overestimation of area and underestimation of the area tend to converge gives the accurate measure of the limit. Later Gamma functions are explained and show clearly how they can be opened up to be used. Later still is Fourier analysis is explained in a rather pedestrian manner.* SummaryIt’s a rewarding book to read and enjoy, you see it from both the set theory and analysis aspects, then calculus runs on rails on top of these aspects and its a wonderfully clear exploration.

⭐nur sind leider keine Lösungen für die Aufgaben erhältlich. Im Konzept dieses Buches sind eingestreute Aufgaben essentiell, d.h. man beweist wichtige Zwischenschritte selbst. Gelingt die Lösung einer solchen Aufgabe nicht, muss man mit einer Lücke leben. Erfahrene Mathematiker werden keine Probleme mit den Aufgaben haben, aber das Buch ist ganz bewusst für die Anfänger geschrieben. Gerade für diese Gruppe wäre ein Lösungsband (oder wenigstens eine entsprechende Internetseite) unverzichtbar.

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⭐L’amateur éclairé que je suis sur ce sujet a trouvé les quelques chapitres que j’ai lu intéressants et instructifs.Pour tous ceux qui veulent bien comprendre la signification de la mesurabilité et donc de la non-mesurabilité, ils y trouveront matière dans ce livre.

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