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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

Absolute continuity of functions

Definition

Properties

Absolute continuity of measures

Relation between the two notions of absolute continuity

Singular measures

See also

References

Absolute continuity of functions

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 ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint sub-intervals [''x''_''k'', ''y''_''k''] of ''I'', ''k'' = 1, 2, ..., ''n'' satisfies
:$sum\_\{k=1\}^\{n\}\; left|\; y\_k\; -\; x\_k$
ight| < delta
then
:$sum\_\{k=1\}^\{n\}\; d\; left(\; f(y\_k),\; f(x\_k)$
ight) < arepsilon.
The collection of all absolutely continuous functions from ''I'' into ''X'' is denoted AC(''I''; ''X'').
A further generalisation is the space AC^''p''(''I''; ''X'') of curves ''f'' : ''I'' → ''X'' such that
:$d\; left(\; f(s),\; f(t)$
ight) leq int_{s}^{t} m( au) , mathrm{d} au mbox{ for all } [s, t] subseteq I
for some ''m'' in the ''L''^''p'' space ''L''^''p''(''I''; 'R').

Properties

★ 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
::
eq 0 end{cases}
: on a finite interval containing the origin, or the function $f(x)=x^2$ 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 $lambda(L)=0$, it holds that $lambda(f(L))=0$, where $lambda$ 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'' ∈ AC^''p''(''I''; ''X''), the metric derivative of ''f'' exists for ''λ''-almost all times in ''I'', and the metric derivative is the smallest ''m'' ∈ ''L''^''p''(''I''; 'R') such that
::$d\; left(\; f(s),\; f(t)$
ight) leq int_{s}^{t} m( au) , mathrm{d} au mbox{ for all } [s, t] subseteq I.

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. It is written as "μ << ν". In symbols:
:$mu\; ll$
u iff left(
u(A) = 0 implies mu (A) = 0
ight).
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'' = ''d''μ/''d''ν, such that for any ν-measurable set ''A'' we have
:$mu(A)=int\_A\; f,d$
u.

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.

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.

See also

★ Singular measure

References

★ Gradient Flows in Metric Spaces and in the Space of Probability Measures, Ambrosio, L., Gigli, N. & Savaré, G., , , ETH Zürich, Birkhäuser Verlag, Basel, 2005, ISBN 3-7643-2428-7

★ Real Analysis, , H.L., Royden, Collier Macmillan, 1968, ISBN 0-02-979410-2