View source for Talk:Absolute continuity

I moved some portions of the old article in [[Signed measure]]. I have not had
the time to add all Mathscinet and Zentralblatt references. [[User:Camillo.delellis|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
--[[User:Jjg|Jjg]] 12:57, 30 July 2012 (CEST)

: Why not? --[[User:Boris Tsirelson|Boris Tsirelson]] 13:21, 30 July 2012 (CEST)
: Fine by me [[User:Camillo.delellis|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... [[User:Camillo.delellis|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). [[User:Camillo.delellis|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? --[[User:Boris Tsirelson|Boris Tsirelson]] 13:06, 10 August 2012 (CEST)

:::: I see: the problem appears when the character "&gt;" appears somewhere after the "&lt;" character in the equation of Theorem 1.
:::: 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)
:::: Some more explanation: if you look at the wikitext of the "References" section of the ''original'' article, you see
::::: &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;
:::: 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)
::::: [//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)