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absolutely continuous function
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(Definition)
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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 $[a,b]$ be a closed bounded interval of
. Then a function
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 (Fundamental theorem of calculus for the Lebesgue integral) Let
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
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.)
See [2,3] for proof.
See also [1], and [4] for a discussion about different proofs.
- 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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"absolutely continuous function" is owned by matte. [ full author list (3) ]
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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 14828 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) |
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Pending Errata and Addenda
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