Absolute continuity

Absolute continuity of real functions

In mathematics, a real-valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [x_k, y_k], k = 1, ..., n satisfies

<math>\sum_{k=1}^n (y_k-x_k)<\delta<math>

then

<math>\sum_{k=1}^n\left|f(y_k)-f(x_k)\right|<\varepsilon.<math>

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.

Absolute continuity of measures

If μ and ν are measures on the same measure space (or, more precisely, on the same sigma-algebra) then μ is absolutely continuous with respect to ν if μ(A) = 0 for every set A for which ν(A) = 0. One writes "μ << ν".

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, i.e., a measurable function f taking values in [0,∞], denoted by f = dμ/dν, such that for any measurable set A we have

<math>\mu(A)=\int_A f\,d\nu.<math>

The connection between absolute continuity of real functions and absolute continuity of measures

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

<math>F(x)=\mu((-\infty,x])<math>

is an absolutely continuous real function.