PlanetMath: absolutely continuous function

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absolutely continuous function

(Definition)

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

Definition Suppose be a closed bounded interval of $\mathbbmss{R}$ . Then a function $f\colon [a,b]\to\mathbbmss{C}$ is absolutely continuous on , 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 such that
$\displaystyle \sum_{i=1}^n (b_i-a_i)< \delta,$
then
$\displaystyle \sum_{i=1}^n \vert f(b_i)-f(a_i)\vert< \varepsilon.$

Theorem 1 (Fundamental theorem of calculus for the Lebesgue integral) Let $f\colon [a,b] \to \mathbbmss{C}$ be a function. Then 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 $\displaystyle \int_a^b \vert g\vert< \infty$ ), such that

$\displaystyle f(x) = f(a) + \int_a^x g(t) dt$

for all $x\in[a,b]$ . What is more, if and are as above, then is differentiable almost everywhere and almost everywhere. (Above, both integrals are Lebesgue integrals.)

See [2,3] for proof.

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

Bibliography

1: Wikipedia, entry on Absolute 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.

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Also defines:

fundamental theorem of calculus for the Lebesgue integral

Attachments:: absolutely continuous on versus absolutely continuous on $[\varepsilon, 1]$ for every $\varepsilon >0$ (Example) by Wkbj79

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Cross-references: proof, integrals, almost everywhere, differentiable, open intervals, disjoint, collection, finite, function, bounded interval, closed, Lebesgue integral, fundamental theorem of calculus
There are 8 references to this entry.

This is version 10 of absolutely continuous function, born on 2005-05-26, modified 2006-10-14.
Object id is 7116, canonical name is AbsolutelyContinuousFunction2.
Accessed 10508 times total.

Classification:

AMS MSC:	26B30 (Real functions :: Functions of several variables :: Absolutely continuous functions, functions of bounded variation)
	26A46 (Real functions :: Functions of one variable :: Absolutely continuous functions)

Pending Errata and Addenda

None.[ View all 1 ]

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