Absolute continuity

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.

Contents 1 Absolute continuity of functions 1.1 Definition 1.2 Properties 2 Absolute continuity of measures 3 Relation between the two notions of absolute continuity 4 Singular measures 5 Examples 6 References

Absolute continuity of functions

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

then

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

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

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 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

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:

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

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.

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

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.

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.

Examples

The following functions are continuous everywhere but not absolutely continuous:

• the Cantor function;
• the function

on a finite interval containing the origin;

• the function ƒ(x) = x ² on an unbounded interval.

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.