In mathematics, the relationship between the two central operations of calculus, differentiation and integration, stated by fundamental theorem of calculus in the framework of Riemann integration, is generalized in several directions, using Lebesgue integration and absolute continuity. 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.
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It may happen that a continuous function f is differentiable almost everywhere on [0,1], its derivative f ′ is Lebesgue integrable, and nevertheless the integral of f ′ differs from the increment of f. For example, this happens for the Cantor function, which means that this function is not absolutely continuous. Absolute continuity of functions is a smoothness property which is stricter than continuity and uniform continuity.
Let I be an interval in the real line R. A function f: I → R 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[1]
then
The collection of all absolutely continuous functions on I is denoted AC(I).
The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:[2]
If these equivalent conditions are satisfied then necessarily 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.
The following functions are continuous everywhere but not absolutely 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
then
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[8]
for some m in the Lp space Lp(I).
A measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure λ (in other words, dominated by λ) if μ(A) = 0 for every set A for which λ(A) = 0. This is written as “μ << λ”.
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 Lebesgue measure is meant.
The same holds for Rn for all n=1,2,3,...
The following conditions on a finite measure μ on Borel subsets of the real line are equivalent:[10]
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 g almost everywhere. Such a function is called Radon-Nikodym derivative, or density, of the absolutely continuous measure μ.
Equivalence between (1), (2) and (3) holds also in Rn for all n=1,2,3,...
Thus, the absolutely continuous measures on Rn are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have probability density functions.
If μ and ν are two measures on the same measurable space then μ is said to be absolutely continuous with respect to ν, or dominated by ν if μ(A) = 0 for every set A for which ν(A) = 0.[11] This is written as “μ ν”. In symbols:
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[12] states that if μ is absolutely continuous with respect to ν, and both measures are σ-finite, then μ has a density, or "Radon-Nikodym derivative", with respect to ν, which means that there exists a ν-measurable function f taking values in [0, +∞], denoted by f = dμ⁄dν, such that for any ν-measurable set A we have
Via Lebesgue's decomposition theorem,[13] every measure can be decomposed into the sum of an absolutely continuous measure and a singular measure. See singular measure for examples of measures that are not absolutely continuous.
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 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.
If the absolute continuity holds then the Radon-Nikodym derivative of μ is equal almost everywhere to the derivative of F.[14]
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.[15] The relation between the two notions of absolute continuity still holds.[16]