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In mathematics, absolute continuity is a smoothness property which is stricter than continuity and uniform continuity. Both absolute continuity of functions and absolute continuity of measures are defined.

[edit] Absolute continuity of functions

[edit] Definition

Let (X, d) be a metric space and let I be an interval in the real line R. A function f: I → X is absolutely continuous on I if for every positive number $ε$ , there is a positive number $δ$ such that whenever a (finite or infinite) sequence of pairwise disjoint sub-intervals [x_k, y_k] of I satisfies

$\sum_{k} \left| y_k - x_k \right| < \delta$

then

$\sum_{k} d \left( f(y_k), f(x_k) \right) < \epsilon.$

The collection of all absolutely continuous functions from I into X is denoted AC(I; X).

A further generalization is the space AC^p(I; X) of curves f: I → X such that

$d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I$

for some m in the L^p space L^p(I; R).

[edit] Properties

The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous.

If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous.

Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.

If f: [a,b] → X is absolutely continuous, then it is of bounded variation on [a,b].

If f: [a,b] → R is absolutely continuous, then it has the Luzin N property (that is, for any $L \subseteq [a,b]$ such that $λ(L) = 0$ , it holds that $λ(f (L)) = 0$ , where $λ$ stands for the Lebesgue measure on R).

If f: I → R is absolutely continuous, then f has a derivative almost everywhere, the derivative is Lebesgue integrable, and its integral is equal to the increment of f.

f: I → R is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property.

For f ∈ AC^p(I; X), the metric derivative of f exists for λ-almost all times in I, and the metric derivative is the smallest m ∈ L^p(I; R) such that

$d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I.$

[edit] Absolute continuity of measures

If μ and ν are two measures on the same measurable space then μ is said to be absolutely continuous with respect to ν, or dominated by ν if μ(A) = 0 for every set A for which ν(A) = 0. This is written as “μ ≪ ν”. In symbols:

$\mu \ll \nu \iff \left( \nu(A) = 0\ \Rightarrow\ \mu (A) = 0 \right).$

Absolute continuity of measures is reflexive and transitive, but is not antisymmetric, so it is a preorder rather than a partial order. Instead, if μ ≪ ν and ν ≪ μ, the measures μ and ν are said to be equivalent. Thus absolute continuity induces a partial ordering of such equivalence classes.

If μ is a signed or complex measure, it is said that μ is absolutely continuous with respect to ν if its variation |μ| satisfies |μ| ≪ ν; equivalently, if every set A for which ν(A) = 0 is μ-null.

The Radon–Nikodym theorem states that if μ is absolutely continuous with respect to ν, and ν is σ-finite, then μ has a density, or "Radon-Nikodym derivative", with respect to ν, which implies that there exists a ν-measurable function f taking values in [0, +∞], denoted by f = ^dμ⁄_dν, such that for any ν-measurable set A we have

$\mu(A) = \int_A f \, \mathrm{d} \nu.$

In most applications, if a measure on n-dimensional Euclidean space Rⁿ is simply said to be absolutely continuous — without specifying with respect to which other measure it is absolutely continuous — then absolute continuity with respect to Lebesgue measure is meant. Since Rⁿ is σ-finite with respect to Lebesgue measure, the absolutely continuous measures on Rⁿ are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have probability density functions.

[edit] Relation between the two notions of absolute continuity

A measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function

$F(x)=\mu((-\infty,x])$

is locally an absolutely continuous real function. In other words, a function is locally absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure.

[edit] Singular measures

Via Lebesgue's decomposition theorem, every measure can be decomposed into the sum of an absolutely continuous measure and a singular measure. See singular measure for examples of non-(absolutely continuous) measures.

[edit] Examples

The following functions are continuous everywhere but not absolutely continuous:

the Cantor function;
the function

$f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x \sin(1/x), & \mbox{if } x \neq 0 \end{cases}$

on a finite interval containing the origin;

the function ƒ(x) = x² on an unbounded interval.

[edit] References

Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 3-7643-2428-7.
Royden, H.L. (1968). Real Analysis. Collier Macmillan. ISBN 0-02-979410-2.

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