Talk:Absolute continuity

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. add references and improve lead--Cronholm144 00:57, 13 May 2007 (UTC)

Is there an example of a uniformly continuous function that is not absolutely continous? Albmont 17:43, 2 January 2007 (UTC)

Yes. The Cantor function, when restricted to the compact interval [0, 1], is a continuous function defined on a compact set, and is therefore uniformly continuous. However, is it not absolutely continuous, as the Cantor distribution is not absolutely continuous with respect to Lebesgue measure. Sullivan.t.j 18:10, 2 January 2007 (UTC)

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I'd say we're missing the alternative characterisation of absolute continuity of measures here, the epsilon-delta one... Anyone can put it on? 189.177.62.204 01:21, 31 May 2007 (UTC)

That's not missing; it's in the article. Michael Hardy 01:25, 31 May 2007 (UTC)

I don't see it! There is an epsilon-delta definiton, but that one is for functions, not for measures. The one I mean is of the sort of: mu is abs. cont. w.r.t. nu if for every epsilon>0 there is a delta>0 such that if a set A satisfies mu(A)<delta then it satisfies nu(A)<epsilon. 189.177.58.19 22:21, 5 June 2007 (UTC)

Oh, OK. Go ahead and put it in. (But write either ν(A) < ε or or the like rather than nu(A)<epsilon.) Maybe I'll put it there if you don't, after I check a couple of sources....) Michael Hardy 22:29, 5 June 2007 (UTC)

you probably mean ν << μ iff for every ε > 0 there is a δ > 0 such that if a set A satisfies μ(A)< δ then it satisfies ν(A) < ε. the (<=) part is obvious. some finiteness assumption seems to be needed on ν, for a short proof of the converse:

suppose the ε-δ condition doesn't hold. so we have some ε and a sequence of sets A_n where ∑ μ(A_n) < ∞ and for every n, ν(A_n) > ε. Take the decreasing sequence B_m = A_m ∪ A_m+1 ... . Then μ(∩ B_m) = 0 but, if ν is finite, ν(∩ B_m) = lim ν(B_m) ≥ ε.

is finiteness necessary? Mct mht 09:23, 25 July 2007 (UTC)

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Should I add some information on so-called AC* functions (absolutely continuous in the narrow sense)? (in definition the value |f(x_k)-f(y_k)| is replaced by osc_{[x_k,y_k]}f )? Or it should be in a different article? And what about ACG and ACG* functions? Probably this would require considering absolute continuity of functions on an arbitrary set E in R, instead of an interval. --a_dergachev (talk) 09:25, 14 February 2008 (UTC)