Absolute continuity

In mathematics, one may talk about absolute continuity of functions and absolute continuity of measures, and these two notions are closely connected.

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 so that whenever a sequence of pairwise disjoint sub-intervals [x_k, y_k] of I, k = 1, 2, ..., n satisfies

$\sum _{k=1}^{n}\left|y_{k}-x_{k}\right|<\delta$

then

$\sum _{k=1}^{n}d\left(f(y_{k}),f(x_{k})\right)<\varepsilon .$

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

A further generalisation 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).

Properties

• The sum, difference and product of two absolutely continuous functions are also absolutely continuous.

• If an absolutely continuous function 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.

• The Cantor function is continuous everywhere but not absolutely continuous; as is 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, or the function $f(x)=x^{2}$ on an infinite interval.

• 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 $L\subseteq [a,b]$ that $\lambda (L)=0$, it holds that $\lambda (f(L))=0$, where $\lambda$ stands for the Lebesgue measure on R).

• If f : I → R is absolutely continuous, then f has a derivative almost everywhere.

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

• 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

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

Absolute continuity of measures

If μ and ν are measures on the same measure space (or, more precisely, on the same σ-algebra) then μ is said to be absolutely continuous with respect to ν if μ(A) = 0 for every set A for which ν(A) = 0. It is written as "μ << ν". In symbols:

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

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

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

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.

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.

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.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Royden, H.L. (1968). Real Analysis. Collier Macmillan. ISBN 0-02-979410-2.