From: Hubert HOLIN Subject: [Q] *elementary* proof of the Lebesgue differentiation theorem. Date: Thu, 03 Aug 2000 00:26:19 +0200 Newsgroups: sci.math.research Paris (U.E.), le 03/08/2000 Hi I am looking for as elementary a proof as possible of the Lebesgue differentiation theorem (a bounded variations numerical function of one real variable is almost everywhere differentiable). The two contenders I have found so far are that of Riesz & Sz.-Nagy (in "Lecons d'Analyse Fonctionnelle"), which is rather a pain to expose in detail without the artful handwaving of the original, and Donald Austin's article "A Geometric Proof of Lebesgue differentiation theorem", Proc. A.M.S., vol. 16, 1965. The problem with the later is that I absolutely do not understand it... Apart from some Vitali-style considerations, there is at the hart of the proof an implication ("hence lf is infinite"), which I can in no way reconcile with what precedes it. I would love to understand what obvious thing I am missing! Are there references to other elementary proofs of that theorem? With thanks Hubert Holin Hubert.Holin@Bigfoot.com ============================================================================== From: artfldodgr@my-deja.com Subject: Re: [Q] *elementary* proof of the Lebesgue differentiation theorem. Date: 3 Aug 2000 15:00:03 -0500 Newsgroups: sci.math.research Summary: [missing] In article <3988A005.8D37D8BA@club-internet.fr>, Hubert.Holin@bigfoot.com, Hubert.Holin@meteo.fr wrote: > The two contenders I have found so far are that of Riesz & Sz.-Nagy > (in "Lecons d'Analyse Fonctionnelle"), which is rather a pain to expose > in detail without the artful handwaving of the original, and Donald > Austin's article "A Geometric Proof of Lebesgue differentiation > theorem", Proc. A.M.S., vol. 16, 1965. > > The problem with the later is that I absolutely do not understand > it... Apart from some Vitali-style considerations, there is at the hart > of the proof an implication ("hence lf is infinite"), which I can in no > way reconcile with what precedes it. I would love to understand what > obvious thing I am missing! Austin shows that there is a positive number b (= (1/6)Leb(E)*(sqrt{1+a^2}-1)) such that if p is any polygonal approximation of f, then there is a second polygonal approximation q with length(q) >= length(p) + b. [lenght(q) = length of the graph of q, etc.] The length L of the graph of f is the sup of the lengths of the polygonal approximations of f. Therefore, L >= L + b. As 0<= L <= infinity, we can only conclude that L= infinity, contradicting the assumed bounded variation of f. --A. Sent via Deja.com http://www.deja.com/ Before you buy.