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In mathematics, one may talk about absolute continuity of functions and absolutely continuity of measures, and these two notions are closely connected.
Absolute continuity of functionsEdit
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, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint sub-intervals [xk, yk] of I, k = 1, 2, ..., n satisfies
The collection of all absolutely continuous functions from I into X is denoted AC(I; X).
A further generalisation is the space ACp(I; X) of curves f : I → X such that
for some m in the Lp space Lp(I; R).
- 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
- on a finite interval containing the origin, or the function 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 that , it holds that , where 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 ∈ ACp(I; X), the metric derivative of f exists for λ-almost all times in I, and the metric derivative is the smallest m ∈ Lp(I; R) such that
Absolute continuity of measuresEdit
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. It 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.
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
The connection between absolute continuity of real functions and absolute continuity of measuresEdit
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.
has the Dirac delta distribution as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure is not absolutely continuous with respect to Lebesgue measure , nor is absolutely continuous with respect to : but ; if is any open set not containing 0, then but .
- 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.
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