absolutely continuous function

is the precise condition one needs to impose in order for the fundamental theorem of calculus to hold for the Lebesgue integral.

Definition Suppose $[a,b]$ be a closed bounded interval of $\mathbbmss{R}$ . Then a function $f\colon[a,b]\to\mathbbmss{C}$ is absolutely continuous on $[a,b]$ , if for any $\varepsilon>0$ , there is a $\delta>0$ such that the following condition holds:

( $\ast$ )

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

$\sum_{i=1}^{n}|f(b_{i})-f(a_{i})|<\varepsilon.$

Theorem 1 ().

Let $f\colon[a,b]\to\mathbbmss{C}$ 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 $g\colon(a,b)\to\mathbbmss{C}$ 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 $f^{\prime}=g$ almost everywhere. (Above, both integrals are Lebesgue integrals.)

See [2, 3] for proof.

See also [1], and [4] for a discussion about different proofs.

References

1 Wikipedia, entry on http://en.wikipedia.org/wiki/Absolute_continuityAbsolute continuity.
2 F. Jones, Lebesgue Integration on Euclidean Spaces, Jones and Barlett Publishers, 1993.
3 C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, 2nd ed., Academic Press, 1990.
4 D. B’arcenas, The Fundamental Theorem of Calculus for Lebesgue Integral, Divulgaciones Matemáticas, Vol. 8, No. 1, 2000, pp. 75-85.

Title	absolutely continuous function
Canonical name	AbsolutelyContinuousFunction
Date of creation	2013-03-22 15:18:47
Last modified on	2013-03-22 15:18:47
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	13
Author	matte (1858)
Entry type	Definition
Classification	msc 26B30
Classification	msc 26A46
Related topic	SingularFunction
Related topic	AbsolutelyContinuous
Defines	fundamental theorem of calculus for the Lebesgue integral