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 be a closed bounded interval of .
Then a function is
absolutely continuous on ,
if for any , there is a such that the following
condition holds:
- ()
Theorem 1 ().
Let be a function. Then is absolutely continuous if and only if there is a function (i.e. a with ), such that
for all .
What is more, if and are as above, then is differentiable
almost everywhere and
almost everywhere. (Above, both integrals are Lebesgue integrals.)
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 |