View source for Absolute continuity

{{MSC|28A33}} (Absolute continuity of measures)

{{MSC|26A46}} (Absolute continuity of functions)

[[Category:Classical measure theory]]

{{TEX|done}}

===Absolute continuity of the Lebesgue integral===

Describes a property of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\lambda$ on the $n$-dimensional
euclidean space and let $f\in L^1 (\mathbb R^n, \lambda)$. Then for every $\varepsilon>0$ there is a $\delta>0$ such that
\[
\left|\int_A f (x) \rd\lambda (x)\right| < \varepsilon \qquad \mbox{for every measurable set}\, A \mbox{ with } \lambda (A)< \delta\, .
\]
This property can be generalized to measures $\mu$ on a $\sigma$-algebra $\mathcal{B}$ of subsets of a space $X$ and
to functions $f\in L^1 (X, \mu)$ (cp. with Theorem 12.34 of {{Cite|HS}}).

===Absolute continuity of measures===
A concept in measure theory (see also [[Absolutely continuous measures]]). If $\mu$ and $\nu$ are two measures on a [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of 
subsets of $X$, we say that $\nu$ is absolutely continuous with respect to $\mu$ if $\nu (A) =0$ for
any $A\in\mathcal{B}$ such that $\mu (A) =0$ (cp. with Defininition 2.11 of {{Cite|Ma}}). This definition can be generalized to [[Signed measure|signed measures]] $\nu$
and even to vector-valued measures $\nu$. Some authors generalize it further to vector-valued $\mu$'s: in that case the absolute continuity of $\nu$ with respect to $\mu$ amounts to the requirement that $\nu (A) = 0$ for any $A\in\mathcal{B}$ such that $|\mu| (A)=0$, where $|\mu|$ is the [[Signed measure|total variation]] of $\mu$ (see for instance Section 30 of {{Cite|Ha}}).

The [[Radon-Nikodym theorem]] (see Theorem B, Section 31 of {{Cite|Ha}}) characterizes the absolute continuity of $\nu$ with respect to $\mu$ with
the existence of a function $f\in L^1 (\mu)$ such that $\nu = f \mu$, i.e. such that 
\[
\nu (A) = \int_A f\, \rd\mu \qquad \text{for every } A\in\mathcal{B}.
\]
A corollary of the Radon-Nikodym, the [[Jordan decomposition (of a signed measure)|Jordan decomposition Theorem]], characterizes signed measures
as differences of nonnegative measures. We refer to [[Signed measure]] for more on this topic (see also [[Hahn decomposition]]).

===Absolute continuity of a function===
A function $f:I\to \mathbb R$, where $I$ is an interval of the real line, 
is said absolutely continuous if for every $\varepsilon> 0$ 
there is $\delta> 0$ such that, for every finite collection of pairwise disjoint intervals $(a_1,b_1), (a_2,b_2), \ldots , (a_n,b_n) \subset I$ with 
$\sum_i (b_i-a_i) <\delta$, we have
\[
\sum_i |f(b_i)-f (a_i)| <\varepsilon
\]
(see Section 4 in Chapter 5 of {{Cite|Ro}}).

An absolutely continuous function is always continuous. Indeed, if the interval of definition is open,
then the absolutely continuous function has a continuous extension to its closure, which is itself
absolutely continuous.
A continuous function might not be absolutely continuous, even if the interval $I$ is compact.
Take for instance the function $f:[0,1]\to \mathbb R$ such that $f(0)=0$ and $f(x) = x \sin x^{-1}$ for
$x>0$. The space of absolutely continuous (real-valued) functions is a vector space. 

A characterization of absolutely continuous functions on an interval might be
given in terms of [[Sobolev space|Sobolev spaces]]: a continuous function $f:I\to \mathbb R$ is absolutely continuous
if and only its [[Generalized derivative|distributional derivative]] is an $L^1$ function, cp. with Theorem 1 in Section 4.9 of {{Cite|EG}} (if $I$ is
bounded, this is equivalent to require $f\in W^{1,1} (I)$). Vice versa, 
for any function with $L^1$ distributional derivative there is an absolutely continuous representative, i.e.
an absolutely continuous $\tilde{f}$ such that $\tilde{f} = f$ a.e. (cp. again with {{Cite|EG}}). The latter statement
can be proved using the absolute continuity of the Lebesgue integral.

An absolutely continuous function is differentiable almost everywhere and its pointwise derivative
coincides with the generalized one. The fundamental theorem of calculus holds for absolutely continuous
functions, i.e. if we denote by $f'$ its pointwise derivative, we then have
\begin{equation}\label{e:fundamental}
f (b)-f(a) = \int_a^b f' (x)\rd x \qquad \forall a<b\in I.
\end{equation}
In fact this is yet another characterization of absolutely continuous functions (see Theorem 13 and Corollary 11 of Section 4 in Chapter 5 of {{Cite|Ro}}).

The differentiability almost everywhere does not imply the absolute continuity: a notable
example is the [[Cantor ternary function]] or "Devil's staircase" (see Problem 46 in Chapter 2 of {{Cite|Ro}}). Though such function is differentiable
almost everywhere, it fails to satisfy \ref{e:fundamental} since the derivative vanishes almost everywhere but the function is not constant, cp. with Problems 11 and 12 of Chapter 5 in {{Cite|Ro}} (indeed the generalized derivative
of the Cantor ternary function is a measure which is not absolutely continuous with respect to
the Lebesgue measure, see {{Cite|AFP}}).

It follows from \ref{e:fundamental} that an absolutely continuous function maps a set of (Lebesgue) measure zero into a set of measure zero (i.e. it has the [[Luzin-N-property]]), and a (Lebesgue) measurable set into a measurable set. Any continuous [[Function of bounded variation|function of bounded variation]] which maps each set of measure zero into a set of measure zero is absolutely continuous (this follows, for instance, from the [[Radon-Nikodym theorem]]). Any absolutely continuous function can be represented as the difference of two absolutely continuous non-decreasing functions.

====Metric setting====
This notion can be easily generalized when the target of the function is a [[Metric space|metric space]] $(X,d)$. In that case the function $f:I\to X$ is absolutely continuous if for every positive $\varepsilon$ there is a positive $\delta$ such that 
for any $a_1<b_1\leq a_2<b_2 \leq \ldots \leq a_n<b_n \in I$ with $\sum_i |a_i -b_i| <\delta$, we have
\[
\sum_i d (f (b_i), f(a_i)) <\varepsilon\, .
\]
The  absolute continuity guarantees the uniform continuity. As for real valued functions, there is a characterization through an appropriate  notion of derivative.

'''Theorem 1'''
A continuous function $f$ is absolutely continuous if and only if there is a function $g\in L^1_{loc} (I, \mathbb R)$ such that
\begin{equation}\label{e:metric}
d (f(b), f(a))\leq \int_a^b g(t)\, dt \qquad \forall a<b\in I\,
\end{equation}
(cp. with {{Cite|AGS}}). This theorem motivates the following

'''Definition 2'''
If  $f:I\to X$ is absolutely continuous and $I$ is a closed interval, the metric  derivative of $f$ is the function $g\in L^1$ with the smallest $L^1$ norm  such that \ref{e:metric} holds (cp. with {{Cite|AGS}}).

The definition can be easily generalized to more general domains of definition. Observe also that, if $X$ is the standard Euclidean space $\mathbb R^k$, then the metric derivative of $f$ is the ''norm'' of the classical derivative.

===References===
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|-
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|}