Lebesgue Integration On Euclidean Space, (Jones and Bartlett Books in Mathematics)

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.

* 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.

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'sReal & Complex Analysis. 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!

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.

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