Talk:Absolute continuity

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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 $\nu (A)<\varepsilon \,$ 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.

Your proof only works if you have finiteness assumptions on $\nu$, because only if at least one of the $B_{n}$ satisfies $\nu (B_{n})<\infty$, your are allowed to compute $\nu (\bigcap B_{n})$ as $\lim _{n\rightarrow \infty }\nu (B_{n})$!

Let me give a counterexample. Let $\lambda$ be Lebesgue measure on the Borel-sets ${\mathcal {B}}$ of the real line. Then define (this is in a sense $\mu :=\infty \cdot \lambda$): $\mu :{\mathcal {B}}\rightarrow \left[0,\infty \right],A\mapsto {\begin{cases}0,&{\text{if }}\lambda \left(A\right)=0\\\infty ,&{\text{if }}\lambda \left(A\right)>0.\end{cases}}$. We shall prove, that $\mu$ is indeed a measure. Clearly $\mu \left(\emptyset \right)=0$. Now let $\left(A_{n}\right)_{n\in \mathbb {N} }\in {\mathcal {B}}^{\mathbb {N} }$ be a sequence of disjoint sets. We consider two cases:

• $\lambda \left(\bigcup _{n=1}^{\infty }A_{n}\right)=0$. Then we have $0\leq \lambda \left(A_{i}\right)\leq \lambda \left(\bigcup _{n=1}^{\infty }A_{n}\right)=0$ for alle $i\in \mathbb {N}$ and therefore by definition of $\mu$: $\sum _{n=1}^{\infty }\mu \left(A_{n}\right)=\sum _{n=1}^{\infty }0=0=\mu \left(\bigcup _{n=1}^{\infty }A_{n}\right).$
• $\lambda \left(\bigcup _{n=1}^{\infty }A_{n}\right)>0$. Then not all $A_{i},i\in \mathbb {N}$ can be of $\lambda$-measure zero, since otherwise we would have $\lambda \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\lambda \left(A_{n}\right)=\sum _{n=1}^{\infty }0=0,$ a contradiction. So let $i_{0}\in \mathbb {N}$ with $\lambda \left(A_{i_{0}}\right)>0$. Then we have again by definition of $\mu$: $\infty =\mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)\geq \sum _{n=1}^{\infty }\mu \left(A_{n}\right)\geq \mu \left(A_{i_{0}}\right)=\infty ,$ and therefore equalitiy.

Furthermore, by definition of absolute continuity and by definition of $\mu$ (we have $\mu \left(A\right)=0$ if $\lambda \left(A\right)=0$ for $A\in {\mathcal {B}}$), we see that $\mu$ is absolutely continuous with respect to $\lambda$. We will now prove that for $\varepsilon :=1>0$ there is no $\delta >0$ such that $\mu \left(A\right)<\varepsilon =1$ if $\lambda \left(A\right)<\delta$. To this end let $\delta >0$ be arbitrary and choose $A_{\delta }:=\left(-{\frac {\delta }{4}},{\frac {\delta }{4}}\right)\in {\mathcal {B}}$. Then $0<\lambda \left(A_{\delta }\right)={\frac {\delta }{2}}<\delta$, but $\mu \left(A_{\delta }\right)=\infty \geq 1=\varepsilon$.

What we have to assume at least to make the statement true (and your proof work), is that $\mu (A)<\infty$ whenever $\lambda (A)<\infty$ or something in that direction. --Phoemuex (talk) 09:19, 11 April 2010 (UTC)

You are right; I was not careful. Two corrections are made. Boris Tsirelson (talk) 16:30, 11 April 2010 (UTC)

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)

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This page ought to have some mention of the Lebesgue version of the Fundamental Theorem of Calculus. To my mind that's a major reason for being interested in absolutely continuous functions. 72.25.102.62 (talk) 01:58, 4 September 2008 (UTC)

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Is the sentence "This relationship is commonly characterized ...." in the first paragraph correct (English)? I can't understand what does it means. — Preceding unsigned comment added by 82.136.67.75 (talk) 07:51, 15 December 2019 (UTC)

"relationship between differentiation and integration is commonly characterized in the framework of Riemann integration" — I think, it means that, having a continuously differentiable function, you can restore it from its derivative by Riemann integration. Poor English? Do you prefer "described" instead of "characterized"? Propose your formulation. Boris Tsirelson (talk) 11:37, 15 December 2019 (UTC)

Lipschitz vs absolute

I am not too familiar with absolute continuity but it seems a little odd that it should be stronger than Lipschitz, as the lead paragraph currently claims. Could someone add a clarification? Katzmik (talk) 12:09, 27 October 2008 (UTC)

P.S. In fact, I see that under "properties" it says the opposite. The lead should be corrected. Katzmik (talk) 12:11, 27 October 2008 (UTC)

Reverting

Something strange happened to the definition in the last two edits. Reverted. Boris Tsirelson (talk) 08:27, 18 June 2009 (UTC)

Interval I in the definition

Am I correct in assuming that this interval does not have to be finite?

TomyDuby (talk) 04:19, 22 November 2009 (UTC)

Split

This should really be two articles: Absolutely continuous function and Absolutely continuous measure. Currently the article has two self-contained sections about two relatively distinct subjects, both of which would seem to be worthwhile subjects for articles of their own. There seems to be no reason for this other than inertia. The current referencing style makes in unclear which source corresponds to which subject, so the references would reviewed to see which article they should be placed in.--RDBury (talk) 20:11, 18 February 2010 (UTC)

No, I believe, these two notions are very much related (as written in Sect.2.1). The sources could be better; I'll add more sources soon. And they will be sources that do describe both subjects (and their relation). True, the relation becomes weak when we turn to functions with values in metric spaces. This case could be treated in a separate section, for convenience of readers that are interested only in real-valued functions. Boris Tsirelson (talk) 20:11, 3 March 2010 (UTC)

Two books are added. I'll also add some inline citations (but not for metric spaces). Boris Tsirelson (talk) 20:28, 3 March 2010 (UTC)

Explanation re a small edit on 7th March

Today, I made what should have been a very small (one character) edit to the bit on R-N theorem. I apologise that it took me 4 attempts to get mu and nu the correct way around in the text of my accompanying edit summary. Hence the multiple undos and redos whose sole purpose was to correct the edit summary, not the page itself. — Preceding unsigned comment added by 138.40.68.40 (talk) 21:25, 7 March 2013 (UTC)

Absolute continuity of functions

"(These examples are continuous but not uniformly continuous.)" — True. but somewhat misleading: a uniformly continuous function (even on [0,1]) need not be absolutely continuous (try the Cantor function). Boris Tsirelson (talk) 09:07, 3 July 2014 (UTC)

Hi Boris. What you say is correct. I was tryin' to say that these are examples which are not absolutely continuous because they are not uniformly continuous, but that there are other functions (like the Cantor function) that are uniformly continuous but not absolutely continuous. But here's a question for you: What about the function $x\sin(1/x)$ in the domain (0, 1]? Is it uniformly continuous? Absolutely continuous? Eric Kvaalen (talk) 12:41, 4 July 2014 (UTC)

OK with "because". Now, the function $x\sin(1/x)$ on the domain (0, 1] is not absolutely continuous; moreover, it is not of bounded variation. Try $x_{n}=1/((n+0.5)\pi )$; you get the divergent harmonic series. And the derivative of this function is not integrable. On the other hand, it is uniformly continuous, since it can be extended by continuity to [0,1]. Also, the function $x^{1+\varepsilon }\sin(1/x)$ on the domain (0, 1] is absolutely continuous whenever $\varepsilon >0.$ Boris Tsirelson (talk) 16:55, 4 July 2014 (UTC)

And by the way, the "Equivalent definitions" are still equivalent on a bounded (but maybe not closed) interval. Boris Tsirelson (talk) 17:10, 4 July 2014 (UTC)

Yeah. It's interestin' to think that no matter how small δ is, you can find an infinite amount of "variation" in the function $x\sin(1/x).$ in the interval (0, δ)! Eric Kvaalen (talk) 06:15, 6 July 2014 (UTC)

And no wonder. Given a divergent series, no matter how far is its tail, you can find an infinite sum in the tail. Boris Tsirelson (talk) 08:15, 6 July 2014 (UTC)

Undefined term: "locally absolutely continuous"

The explanation in the subsection "Relation between the two notions of absolute continuity" uses the term "locally absolutely continuous", but this term was not defined before. Can someone please add a definition? --Erel Segal (talk) 14:28, 22 August 2014 (UTC)

Really, here "locally" is not needed, since the measure is assumed to be finite. (Otherwise, "locally" is needed, and means: on every bounded interval.) Boris Tsirelson (talk) 15:18, 22 August 2014 (UTC)

Absolute continuity in more than one dimension

The Generalizations section discusses functions whose range is a general metric space, but the domain is still the real line. Is there a standard way to generalize absolute continuity to functions whose domain is a general metric space, or at least a multi-dimensional space? E.g., when does a function F(x,y) on R² considered absolutely continuous? --Erel Segal (talk) 20:01, 23 August 2014 (UTC)

I never saw such a generalization for metric space domain. For R² a straighforward thought is that a cumulative distribution function of an absolutely continuous measure fits. But still, I did not see it.

About the chain of inclusions I wonder: what is meant by differentiability of a function over a compact subset of the real line? Maybe a compact interval is meant (here and "Lipschitz continuous" as well)? Boris Tsirelson (talk) 20:59, 23 August 2014 (UTC)

This makes sense, but you probably know better than I do. I just copied the chain from elsewhere. --Erel Segal (talk) 17:24, 25 August 2014 (UTC)

Inclusion question

It currently reads:

"We have the following chains of inclusions for functions over a compact subset of the real line:

absolutely continuous ⊆ uniformly continuous ⊆ continuous

"

If we are restricting ourselves to compact intervals of the real line then uniformly continuous is equivalent to continuous, correct? What is this trying to say here?

Yes, you are right. And the same happens on Lipschitz_continuity. Not wrong, but rather ridiculous. I'll replace "⊆" with "=" (on both pages). Boris Tsirelson (talk) 15:32, 28 April 2015 (UTC)

What about Cantor function?

It is uniformly continuous but not absolutely continuous. So what? Boris Tsirelson (talk) 19:27, 3 July 2015 (UTC)

Perhaps

absolutely continuous ⊂ uniformly continuous = continuous

is better. I suspect (but do not know) that there are examples to show that some (all?) of the other inclusions given in the lead are strict too. YohanN7 (talk) 14:16, 16 June 2016 (UTC)

As far as I remember, various editors change these things repeatedly, back and forth, here and in "Lipschitz_continuity"; and I do not know how to stop this process. (Similarly, "die" and "dice" are changed repeatedly, back and forth, in some probability articles.) Boris Tsirelson (talk) 15:43, 16 June 2016 (UTC)