Absolute continuity
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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

then

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-Nikdoym 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

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

is an absolutely continuous real function.

The original article can be found at: http://en.wikipedia.org/wiki/Absolute_continuity.
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