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:
and, for a compact interval,
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.
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
The collection of all absolutely continuous functions on is denoted .
The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:
If these equivalent conditions are satisfied, then necessarily any function g as in condition 3. satisfies g = f ′ almost everywhere.
For an equivalent definition in terms of measures see the section Relation between the two notions of absolute continuity.
The following functions are uniformly continuous but not absolutely continuous:
The following functions are absolutely continuous but not α-Hölder continuous:
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 , there is a positive number such that whenever a finite sequence of pairwise disjoint sub-intervals [xk, yk] of I satisfies:
The collection of all absolutely continuous functions from I into X is denoted AC(I; X).
A further generalization is the space ACp(I; X) of curves f: I → X such that:
for some m in the Lp space Lp(I).
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
The following conditions on a finite measure on Borel subsets of the real line are equivalent:
For an equivalent definition in terms of functions see the section Relation between the two notions of absolute continuity.
Any other function satisfying (3) is equal to almost everywhere. Such a function is called Radon–Nikodym derivative, or density, of the absolutely continuous measure
Equivalence between (1), (2) and (3) 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.
When then is said to be dominating
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.
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:
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.
is an absolutely continuous real function. More generally, a function is locally (meaning on every bounded interval) 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 (rather than finite) and F(x) is defined as μ((0,x]) for x > 0, 0 for x = 0, and −μ((x,0]) for x < 0. In this case μ is the Lebesgue–Stieltjes measure generated by F. The relation between the two notions of absolute continuity still holds.