absolutely continuous functionAbsolutelyContinuousFunction22005-05-26 10:34:092006-10-14 00:34:20Definitionfundamental theorem of calculus for the Lebesgue integral% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}\PMlinkescapeword{order}
\PMlinkescapetext{Absolute continuity} is the precise condition one needs to
impose in order for the fundamental theorem of calculus
to hold for the Lebesgue integral.
\PMlinkescapeword{absolutely continuous}
\PMlinkescapeword{property}
{\bf Definition}
Suppose $[a,b]$ be a closed bounded interval of $\R$.
Then a function $f\colon [a,b]\to\C$ is
{\bf absolutely continuous} on $[a,b]$,
if for any $\varepsilon>0$, there is a $\delta>0$ such that the following
condition holds:
\begin{itemize}
\item[($\ast$)] If $(a_1,b_1), \ldots, (a_n,b_n)$ is a finite
collection of disjoint open intervals in $[a,b]$
such that
$$
\sum_{i=1}^n (b_i-a_i)< \delta,
$$
then
$$
\sum_{i=1}^n |f(b_i)-f(a_i)|< \varepsilon.
$$
\end{itemize}
\begin{thm}[\PMlinkescapetext{Fundamental theorem of calculus for the Lebesgue integral}]
Let $f\colon [a,b] \to \C$ be a
function. Then $f$ is absolutely continuous if and only if
there is a function $g\in L^1(a,b)$ (i.e. a $g\colon(a,b)\to \C$ with
$\displaystyle \int_a^b |g|< \infty$), such that
$$
f(x) = f(a) + \int_a^x g(t) dt
$$
for all $x\in[a,b]$.
What is more, if $f$ and $g$ are as above, then $f$ is differentiable
almost everywhere and $f'=g$
almost everywhere. (Above, both integrals are Lebesgue integrals.)
\end{thm}
See \cite{jones,aliprantis} for proof.
See also \cite{wikiabs}, and \cite{barcenas} for a discussion
about different proofs.
\begin{thebibliography}{9}
\bibitem{wikiabs} Wikipedia, entry on
\PMlinkexternal{Absolute continuity}{http://en.wikipedia.org/wiki/Absolute_continuity}.
\bibitem{jones}
F. Jones, \emph{Lebesgue Integration on Euclidean Spaces},
Jones and Barlett Publishers, 1993.
\bibitem{aliprantis}
C.D. Aliprantis, O. Burkinshaw, \emph{Principles of Real Analysis},
2nd ed., Academic Press, 1990.
\bibitem{barcenas} D. B'arcenas,
\emph{The Fundamental Theorem of
Calculus for Lebesgue Integral},
Divulgaciones Matem\'aticas, Vol. 8, No. 1, 2000, pp. 75-85.
\end{thebibliography}