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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, ?/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:
(1) f is absolutely continuous;
(2) f has a derivative f ? almost everywhere, the derivative is Lebesgue integrable, and
for all x on [a,b];
(3) there exists a Lebesgue integrable function g on [a,b] such that
for all x in [a,b].
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.
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 is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous.
Let (X, d) be a metric space and let I be an interval in the real lineR. 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
A measure on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure (in other words, dominated by ) if for every measurable set , implies . This is written as .
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 works for measures on Borel subsets of .
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 A of Lebesgue measure less than ?;
(3) there exists a Lebesgue integrable function g on the real line such that
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 ?.
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 ? and ? are two measures on the same measurable space then ? is said to be absolutely continuous with respect to ?, or dominated by ? if ?(A) = 0 for every set A for which ?(A) = 0. This is written as "??". That is:
If ? is a signed or complex measure, it is said that ? is absolutely continuous with respect to ? if its variation |?| satisfies |?| A for which ?(A) = 0 is ?-null.
The Radon-Nikodym theorem states that if ? is absolutely continuous with respect to ?, and both measures are ?-finite, then ? has a density, or "Radon-Nikodym derivative", with respect to ?, which means that there exists a ?-measurable function f taking values in [0, +?), denoted by f = d?/d?, such that for any ?-measurable set A we have
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 ? is equal almost everywhere to the derivative of F.
More generally, the measure ? is assumed to be locally finite (rather than finite) and F(x) is defined as ?((0,x]) for , 0 for , and -?((x,0]) for . In this case ? is the Lebesgue-Stieltjes measure generated by F.
The relation between the two notions of absolute continuity still holds.
^Royden 1988, Sect. 5.4, page 108; Nielsen 1997, Definition 15.6 on page 251; Athreya & Lahiri 2006, Definitions 4.4.1, 4.4.2 on pages 128,129. The interval is assumed to be bounded and closed in the former two books but not the latter book.
^Equivalence between (1) and (2) is a special case of Nielsen 1997, Proposition 15.5 on page 251 (fails for ?-finite measures); equivalence between (1) and (3) is a special case of the Radon-Nikodym theorem, see Nielsen 1997, Theorem 15.4 on page 251 or Athreya & Lahiri 2006, Item (ii) of Theorem 4.1.1 on page 115 (still holds for ?-finite measures).