Absolute continuity


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 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:
and, for a compact interval,

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 over, x2 over the entire real line, and sin over. 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. This happens for example with the Cantor function.

Definition

Let be an interval in the real line. A function is absolutely continuous on if for every positive number, there is a positive number such that whenever a finite sequence of pairwise disjoint sub-intervals of with satisfies
then
The collection of all absolutely continuous functions on is denoted.

Equivalent definitions

The following conditions on a real-valued function f on a compact interval are equivalent:
  1. f is absolutely continuous;
  2. f has a derivative f almost everywhere, the derivative is Lebesgue integrable, and for all x on ;
  3. there exists a Lebesgue integrable function g on such that 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 and is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue.
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.
  • If an absolutely continuous function f is defined on a bounded closed interval and is nowhere zero then 1/f is absolutely continuous.
  • Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.
  • If f: → R is absolutely continuous, then it is weakly differentiable; conversely if f: → R is weakly differentiable, then it coincides almost everywhere with an absolutely continuous function ; this provides a characterization of Sobolev spaces on intervals of the real line.
  • If f: → R is absolutely continuous, then it is of bounded variation on .
  • 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.
  • 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.
  • 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.

    Examples

The following functions are uniformly continuous but not absolutely continuous:
  • The Cantor function on ;
  • The function: on a finite interval containing the origin.
The following functions are absolutely continuous but not α-Hölder continuous:
  • The function f = xβ on , for any
The following functions are absolutely continuous and α-Hölder continuous but not Lipschitz continuous:
  • The function f = on , for α ≤ 1/2.

    Generalizations

Let 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, there is a positive number such that whenever a finite sequence of pairwise disjoint sub-intervals of I satisfies:
then:
The collection of all absolutely continuous functions from I into X is denoted AC.
A further generalization is the space ACp of curves f: I → X such that:
for some m in the Lp space Lp.

Properties of these generalizations

  • Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.
  • If f: → X is absolutely continuous, then it is of bounded variation on .
  • For f ∈ ACp, the metric derivative of f exists for λ-almost all times in I, and the metric derivative is the smallest m ∈ Lp such that:

    Absolute continuity of measures

Definition

A measure on Borel subsets of the real line is absolutely continuous with respect to the Lebesgue measure if for every -measurable set implies. Equivalently, implies. This condition is written as We say is dominated by
In most applications, if a measure on the real line 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 the Lebesgue measure is meant.
The same principle holds for measures on Borel subsets of

Equivalent definitions

The following conditions on a finite measure on Borel subsets of the real line are equivalent:
  1. is absolutely continuous;
  2. For every positive number there is a positive number such that for all Borel sets of Lebesgue measure less than
  3. There exists a Lebesgue integrable function on the real line such that: for all Borel subsets of the real line.
For an equivalent definition in terms of functions see the section Relation between the two notions of absolute continuity.
Any other function satisfying is equal to almost everywhere. Such a function is called Radon–Nikodym derivative, or density, of the absolutely continuous measure
Equivalence between, and holds also in for all
Thus, the absolutely continuous measures on are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have probability density functions.

Generalizations

If and are two measures on the same measurable space is said to be with respect to if for every set for which This is written as "". That is:
When then is said to be
Absolute continuity of measures is reflexive and transitive, but is not antisymmetric, so it is a preorder rather than a partial order. Instead, if and the measures and are said to be equivalent. Thus absolute continuity induces a partial ordering of such equivalence classes.
If is a signed or complex measure, it is said that is absolutely continuous with respect to if its variation satisfies equivalently, if every set for which is -null.
The Radon–Nikodym theorem states that if is absolutely continuous with respect to and both measures are σ-finite, then has a density, or "Radon-Nikodym derivative", with respect to which means that there exists a -measurable function taking values in denoted by such that for any -measurable set we have:

Singular measures

Via Lebesgue's decomposition theorem, every σ-finite measure can be decomposed into the sum of an absolutely continuous measure and a singular measure with respect to another σ-finite measure. See singular measure for examples of measures that are not absolutely continuous.

Relation between the two notions of absolute continuity

A finite measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function:
is an absolutely continuous real function. More generally, 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.
If absolute continuity holds then the Radon–Nikodym derivative of μ is equal almost everywhere to the derivative of F.
More generally, the measure μ is assumed to be locally finite and F is defined as μ for, 0 for, and −μ for. In this case μ is the Lebesgue–Stieltjes measure generated by F. The relation between the two notions of absolute continuity still holds.