Absolute continuity (measure theory)

In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure. We have the following chains of inclusions for functions over a compact subset of the real line:

absolutely continuous ⊆ uniformly continuous

=

continuous

and, for a compact interval,

continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere.

Absolute continuity of functions

A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over $[0, π /2)$ , x² over the entire real line, and sin(1/x) over (0, 1. But a continuous function f can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the Weierstrass function, which is not differentiable anywhere). Or it may be differentiable almost everywhere and its derivative f ′ may be Lebesgue integrable, but the integral of f ′ differs from the increment of f (how much f changes over an interval). This happens for example with the Cantor function.

Definition

Let $I$ be an interval in the real line $\mathbb {R}$ . A function $f\colon I\to \mathbb {R}$ is absolutely continuous on $I$ if for every positive number $\varepsilon$ , there is a positive number $\delta$ such that whenever a finite sequence of pairwise disjoint sub-intervals $(x_{k},y_{k})$ of $I$ with $x_{k}<y_{k}\in I$ satisfies^[1]

\sum _{k=1}^{N}(y_{k}-x_{k})<\delta

then

\sum _{k=1}^{N}|f(y_{k})-f(x_{k})|<\varepsilon .

The collection of all absolutely continuous functions on $I$ is denoted $\operatorname {AC} (I)$ .

Equivalent definitions

The following conditions on a real-valued function f on a compact interval are equivalent:^[2]

f is absolutely continuous;
f has a derivative f ′ almost everywhere, the derivative is Lebesgue integrable, and $f(x)=f(a)+\int _{a}^{x}f'(t)\,dt$ for all x on ;
there exists a Lebesgue integrable function g on such that $f(x)=f(a)+\int _{a}^{x}g(t)\,dt$ for all x in .

If these equivalent conditions are satisfied, then necessarily any function g as in condition 3. satisfies g = f ′ almost everywhere.

Equivalence between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue.^[3]

For an equivalent definition in terms of measures see the section Relation between the two notions of absolute continuity.

Properties

The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous.^[4]
If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous.^[5]
Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every (globally) Lipschitz-continuous function is absolutely continuous.^[6]
If f: → R is absolutely continuous, then it is of bounded variation on .^[7]
If f: → R is absolutely continuous, then it can be written as the difference of two monotonic nondecreasing absolutely continuous functions on .
If f: → R is absolutely continuous, then it has the Luzin N property (that is, for any $N\subseteq$ such that $\lambda (N)=0$ , it holds that $\lambda (f(N))=0$ , where $\lambda$ stands for the Lebesgue measure on R).
f: I → R is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property. This statement is also known as the Banach-Zareckiǐ theorem.^[8]
If f: I → R is absolutely continuous and g: R → R is globally Lipschitz-continuous, then the composition g ∘ f is absolutely continuous. Conversely, for every function g that is not globally Lipschitz continuous there exists an absolutely continuous function f such that g ∘ f is not absolutely continuous.^[9]

Examples

The following functions are uniformly continuous but not absolutely continuous:

The Cantor function on (it is of bounded variation but not absolutely continuous);
The function: $f(x)={\begin{cases}0,&{\text{if }}x=0\\x\sin(1/x),&{\text{if }}x\neq 0\end{cases}}$ on a finite interval containing the origin.

The following functions are absolutely continuous but not α-Hölder continuous:

The function f(x) = x^β on , for any 0 < β < α < 1

The following functions are absolutely continuous and α-Hölder continuous but not Lipschitz continuous:

The function f(x) = √x on , for α ≤ 1/2.

Generalizations

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 $\epsilon$ , there is a positive number $\delta$ such that whenever a finite sequence of pairwise disjoint sub-intervals of I satisfies: