Lebesgue Integration on Euclidean Space - Revised Mathematics Textbook | Advanced Calculus & Real Analysis Guide for University Students & Researchers

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!