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

$\sum _{k=1}^{n}(y_{k}-x_{k})<\delta$

then

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

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

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

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

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

is an absolutely continuous real function.