Absolute continuity

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

Contents

[edit] Absolute continuity of functions

[edit] 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 \varepsilon, there is a positive number δ such that whenever a sequence of pairwise disjoint sub-intervals [xk, yk] 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 ACp(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 Lp space Lp(I; R).

[edit] 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.
  • 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) = x2 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 λ(L) = 0, it holds that λ(f(L)) = 0, where λ stands for the Lebesgue measure on R).
  • If f : I → R is absolutely continuous, then f has a derivative almost everywhere.
  • f : I → R is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property.
  • 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
d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I.

[edit] 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.

In most applications, if a measure on n-dimensional Euclidean space Rn is simply said to be absolutely continuous — without specifying with respect to which other measure it is absolutely continuous — then absolute continuity with respect to Lebesgue measure is meant. Since Rn is σ-finite with respect to Lebesgue measure, the absolutely continuous measures on Rn are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have probability density functions.

[edit] 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.

[edit] 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.

[edit] Examples

Continuity does not imply absolute continuity; the Cantor function is continuous but not absolutely continuous.

[edit] 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. 
  • Royden, H.L. (1968). Real Analysis. Collier Macmillan. ISBN 0-02-979410-2.