﻿ Absolute continuity

# Absolute continuity

In calculus, the absolute continuity of a function is a tightening of the property of consistency. The term was introduced in 1905 by Giuseppe Vitali and allows a characterization of Lebesgue integrals.

In measure theory, the absolute continuity is a property of moderation. A measure is absolutely continuous with respect to a measure, if any -null set is a null set with respect. The feature makes it possible to characterize such dimensions, which can be represented by a density function; is a close relationship between the two concepts.

## Absolute continuity of real functions

More is, a defined on an interval real-valued function absolutely continuous if for every number there is a number which is so small that for any finite or infinite sequence of pairwise disjoint intervals which are subsets of and the condition

Satisfy the following relationship applies:

Every absolutely continuous function is uniformly continuous and hence in particular continuous. On the other hand, every Lipschitz continuous function is absolutely continuous.

The Cantor function is an example of an everywhere continuous but not absolutely continuous function.

Absolutely continuous functions are almost everywhere differentiable and this derivation is consistent with the weak derivative agreement.

## Absolute continuity of measures

Are and dimensions on the σ - algebra, so is called absolutely continuous (or short: continuous) with respect to, if for all:

You write short and also speaks alternatively assume that the measure dominates.

A measure on the real number line is if and only absolutely continuous with respect to Lebesgue measure on the Borel sets of real numbers if for every finite interval of the restriction

Is an absolutely continuous real function.

## Applications

• In the theory of optimal control is so far demanded that the Lösungstrajektorien are absolutely continuous.
• The Radon - Nikodym theorem states: If the measure is absolutely continuous with respect to a measure and σ - finite, then has a density function, sometimes called the Radon - Nikodym derivative with respect to, ie there is a measurable function, with is known, such that for each measurable quantity:
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