| Version 2 |
Version 1 |
| Absolutele continuity is the precise condition one needs to |
Absolutele continuity is the precise condition one needs to |
| impose in order for the fundamental theorem of calculus |
impose in order for the fundamental theorem of calculus |
| should hold for the Lebesgue integral. |
should hold for the Lebesgue integral. |
|
|
| \PMlinkescapeword{absolutely continuous} |
\PMlinkescapeword{absolutely continuous} |
| \PMlinkescapeword{property} |
\PMlinkescapeword{property} |
| {\bf Definition} |
{\bf Definition} |
| Suppose $[a,b]$ be a closed bounded interval of $\R$. |
Suppose $[a,b]$ be a closed bounded interval of $\R$. |
| Then a function $f\colon [a,b]\to\C$ is |
Then a function $f\colon [a,b]\to\C$ is |
| {\bf absolutely continuous} on $[a,b]$, |
{\bf absolutely continuous} on $[a,b]$, |
| if for any $\varepsilon>0$, there is a $\delta>0$ such that the following |
if for any $\varepsilon>0$, there is a $\delta>0$ such that the following |
| condition holds: |
condition holds: |
| \begin{itemize} |
\begin{itemize} |
| \item[($\ast$)] If $(a_1,b_1), \ldots, (a_n,b_n)$ is a finite |
\item[($\ast$)] If $(a_1,b_1), \ldots, (a_n,b_n)$ is a finite |
| collection of disjoint open intervals in $[a,b]$ |
collection of disjoint open intervals in $[a,b]$ |
| such that |
such that |
| $$ |
$$ |
| \sum_{i=1}^n (b_i-a_i)< \delta, |
\sum_{i=1}^n (b_i-a_i)< \delta, |
| $$ |
$$ |
| then |
then |
| $$ |
$$ |
| \sum_{i=1}^n |f(b_i)-f(a_i)|< \varepsilon. |
\sum_{i=1}^n |f(b_i)-f(a_i)|< \varepsilon. |
| $$ |
$$ |
| \end{itemize} |
\end{itemize} |
|
|
| \begin{thm}[Fundamental theorem of calculus for the Lebesgue integral |
\begin{thm}[Fundamental theorem of calculus for the Lebesgue integral |
| \cite{jones,aliprantis}] |
\cite{jones,aliprantis}] |
| Let $f\colon [a,b] \to \C$ be a |
Let $f\colon [a,b] \to \C$ be a |
| function. Then $f$ is absolutely continuous if and only if |
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 |
there is a function $g\in L^1(a,b)$ (i.e. a $g\colon(a,b)\to \C$ with |
| $\int_{(a,b)} |g|< \infty$), such that |
$\int_{(a,b)} |g|< \infty$), such that |
| $$ |
$$ |
| f(x) = f(a) + \int_a^x g(t) dt |
f(x) = f(a) + \int_a^x g(t) dt |
| $$ |
$$ |
| for all $x\in[a,b]$. |
for all $x\in[a,b]$. |
| What is more, if $f$ and $g$ are as above, then $f$ is differentiable |
What is more, if $f$ and $g$ are as above, then $f$ is differentiable |
| almost everywhere and $f'=g$ |
almost everywhere and $f'=g$ |
| almost everywhere. (Above, both integrals are Lebesgue integrals.) |
almost everywhere. (Above, both integrals are Lebesgue integrals.) |
| \end{thm} |
\end{thm} |
|
|
| See also \cite{wikiabs}. |
See also \cite{wikiabs}. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{wikiabs} Wikipedia, entry on |
\bibitem{wikiabs} Wikipedia, entry on |
| \PMlinkexternal{Absolute continuity}{http://en.wikipedia.org/wiki/Absolute_continuity}. |
\PMlinkexternal{Absolute continuity}{http://en.wikipedia.org/wiki/Absolute_continuity}. |
| \bibitem{jones} |
\bibitem{jones} |
| F. Jones, \emph{Lebesgue Integration on Euclidean Spaces}, |
F. Jones, \emph{Lebesgue Integration on Euclidean Spaces}, |
| Jones and Barlett Publishers, 1993. |
Jones and Barlett Publishers, 1993. |
| \bibitem{aliprantis} |
\bibitem{aliprantis} |
| C.D. Aliprantis, O. Burkinshaw, \emph{Principles of Real Analysis}, |
C.D. Aliprantis, O. Burkinshaw, \emph{Principles of Real Analysis}, |
| 2nd ed., Academic Press, 1990. |
2nd ed., Academic Press, 1990. |
| \end{thebibliography} |
\end{thebibliography} |
