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Difference between revisions of "Talk:Absolute continuity"

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* calligraphic is being used already for $\sigma$-algebras
 
* calligraphic is being used already for $\sigma$-algebras
 
--[[User:Jjg|Jjg]] 12:57, 30 July 2012 (CEST)
 
--[[User:Jjg|Jjg]] 12:57, 30 July 2012 (CEST)
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: Why not? --[[User:Boris Tsirelson|Boris Tsirelson]] 13:21, 30 July 2012 (CEST)
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: Fine by me [[User:Camillo.delellis|Camillo]] 14:08, 30 July 2012 (CEST)
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Between metric setting and References I would like to type the following lines. But for some reason which is misterious to me, any time I try the page comes out a mess... [[User:Camillo.delellis|Camillo]] 10:45, 10 August 2012 (CEST)
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if for every $\varepsilon$ there is a $\delta > 0$ such that,
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for any $a_1<b_1<a_2<b_2<\ldots < a_n<b_n \in I$ with $\sum_i |a_i -b_i| <\delta$, we have
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\[
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\sum_i d (f (b_i), f(a_i)) <\varepsilon\, .
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\]
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The  absolute continuity guarantees the uniform continuity. As for real  valued functions, there is a characterization through an appropriate  notion of derivative.
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'''Theorem 1'''
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A continuous function $f$ is absolutely continuous if and only if there is a function $g\in L^1_{loc} (I, \mathbb R)$ such that
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\begin{equation}\label{e:metric}
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d (f(b), f(a))\leq \int_a^b g(t)\, dt \qquad \forall a<b\in I\,
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\end{equation}
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(cp. with ). This theorem motivates the following
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'''Definition 2'''
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If  $f:I\to X$ is a absolutely continuous and $I$ is compact, the metric  derivative of $f$ is the function $g\in L^1$ with the smalles $L^1$ norm  such that \ref{e:metric} holds (cp. with )
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:: OK, I found a way around. But there must be some bug: it seems that whenever I write the symbol "bigger" then things gets messed up (now even on THIS page). [[User:Camillo.delellis|Camillo]] 10:57, 10 August 2012 (CEST)
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::: But I did not understand what is the problem. Messed up? On this page? Where? And what was the way around? --[[User:Boris Tsirelson|Boris Tsirelson]] 13:06, 10 August 2012 (CEST)
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:::: I see: the problem appears when the character "&gt;" appears somewhere after the "&lt;" character in the equation of Theorem 1.
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:::: Wow! Really, I knew that some characters, including "&lt;", "&gt;", "&amp;" are forbidden in the wikitext, because they are special characters of the HTML language. But I also knew that the wiki software is cute enough, able to interpret these symbols as needed in most cases. Well, you are non-lucky enough to find a bad case. My experiment shows that it is enough to put a space after the "&lt;" character. I do not know whether this is always enough or not. As the last resort, you can write "&amp;lt;" instead of "&lt;" and "&amp;gt;" instead of "&gt;"; this must work always; but I hope that a simpler way round can be found when needed (as you did). --[[User:Boris Tsirelson|Boris Tsirelson]] 15:38, 10 August 2012 (CEST)
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:::: Some more explanation: if you look at the wikitext of the "References" section of the ''original'' article, you see
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::::: &lt;table&gt;&lt;TR&gt;&lt;TD valign="top"&gt;[1]&lt;/TD&gt; &lt;TD  valign="top"&gt;  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)&lt;/TD&gt;
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:::: and so on. Here html tags are used intensively; they start with &lt; and end with &gt;. Now you can see what is the problem for the software... --[[User:Boris Tsirelson|Boris Tsirelson]] 16:29, 10 August 2012 (CEST)
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::::: [//en.wikipedia.org/wiki/Help:HTML_in_wikitext  Maybe this page helps clearifying things]. --[[User:Ulf Rehmann|Ulf Rehmann]] 18:26, 10 August 2012 (CEST)

Latest revision as of 18:26, 10 August 2012

I moved some portions of the old article in Signed measure. I have not had the time to add all Mathscinet and Zentralblatt references. Camillo 22:54, 29 July 2012 (CEST)

Could I suggest using $\lambda$ rather than $\mathcal L$ for Lebesgue measure since

  • it is very commonly used, almost standard
  • it would be consistent with the notation for a general measure, $\mu$
  • calligraphic is being used already for $\sigma$-algebras

--Jjg 12:57, 30 July 2012 (CEST)

Why not? --Boris Tsirelson 13:21, 30 July 2012 (CEST)
Fine by me Camillo 14:08, 30 July 2012 (CEST)

Between metric setting and References I would like to type the following lines. But for some reason which is misterious to me, any time I try the page comes out a mess... Camillo 10:45, 10 August 2012 (CEST)


if for every $\varepsilon$ there is a $\delta > 0$ such that, for any $a_1<b_1<a_2<b_2<\ldots < a_n<b_n \in I$ with $\sum_i |a_i -b_i| <\delta$, we have \[ \sum_i d (f (b_i), f(a_i)) <\varepsilon\, . \] The absolute continuity guarantees the uniform continuity. As for real valued functions, there is a characterization through an appropriate notion of derivative.

Theorem 1 A continuous function $f$ is absolutely continuous if and only if there is a function $g\in L^1_{loc} (I, \mathbb R)$ such that \begin{equation}\label{e:metric} d (f(b), f(a))\leq \int_a^b g(t)\, dt \qquad \forall a<b\in I\, \end{equation} (cp. with ). This theorem motivates the following

Definition 2 If $f:I\to X$ is a absolutely continuous and $I$ is compact, the metric derivative of $f$ is the function $g\in L^1$ with the smalles $L^1$ norm such that \ref{e:metric} holds (cp. with )

OK, I found a way around. But there must be some bug: it seems that whenever I write the symbol "bigger" then things gets messed up (now even on THIS page). Camillo 10:57, 10 August 2012 (CEST)
But I did not understand what is the problem. Messed up? On this page? Where? And what was the way around? --Boris Tsirelson 13:06, 10 August 2012 (CEST)
I see: the problem appears when the character ">" appears somewhere after the "<" character in the equation of Theorem 1.
Wow! Really, I knew that some characters, including "<", ">", "&" are forbidden in the wikitext, because they are special characters of the HTML language. But I also knew that the wiki software is cute enough, able to interpret these symbols as needed in most cases. Well, you are non-lucky enough to find a bad case. My experiment shows that it is enough to put a space after the "<" character. I do not know whether this is always enough or not. As the last resort, you can write "&lt;" instead of "<" and "&gt;" instead of ">"; this must work always; but I hope that a simpler way round can be found when needed (as you did). --Boris Tsirelson 15:38, 10 August 2012 (CEST)
Some more explanation: if you look at the wikitext of the "References" section of the original article, you see
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.R. Halmos, "Measure theory" , v. Nostrand (1950)</TD>
and so on. Here html tags are used intensively; they start with < and end with >. Now you can see what is the problem for the software... --Boris Tsirelson 16:29, 10 August 2012 (CEST)
Maybe this page helps clearifying things. --Ulf Rehmann 18:26, 10 August 2012 (CEST)
How to Cite This Entry:
Absolute continuity. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Absolute_continuity&oldid=27246