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:
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 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[1]
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 [a,b] are equivalent:[2]
f is absolutely continuous;
f has a derivative f ′
almost everywhere, the derivative is Lebesgue integrable, and
for all x on [a,b];
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 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]
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]
If f: [a,b] → R is absolutely continuous, then it is of
bounded variation on [a,b].[7]
If f: [a,b] → R is absolutely continuous, then it can be written as the difference of two monotonic nondecreasing absolutely continuous functions on [a,b].
If f: [a,b] → R is absolutely continuous, then it has the
Luzin N property (that is, for any such that , it holds that , where 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 [0, 1] (it is of bounded variation but not absolutely continuous);
The function:
on a finite interval containing the origin.
The following functions are absolutely continuous but not α-Hölder continuous:
The function f(x) = xβ on [0, c], for any 0 < β < α < 1
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:
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:[10]
If f: [a,b] → X is absolutely continuous, then it is of
bounded variation on [a,b].
For f ∈ ACp(I; X), the
metric derivative of f exists for λ-
almost all times in I, and the metric derivative is the smallest m ∈ Lp(I; R) such that:[11]
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:[12]
is absolutely continuous;
For every positive number there is a positive number such that for all Borel sets of Lebesgue measure less than
There exists a Lebesgue integrable function on the real line such that:
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.
Generalizations
If and are two
measures on the same
measurable space is said to be absolutely continuous with respect to if for every set for which [13] 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 equivalently, if every set for which is -
null.
The
Radon–Nikodym theorem[14] 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,[15] 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 (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.[16]
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.[17] The relation between the two notions of absolute continuity still holds.[18]
Notes
^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 σ-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 σ-finite measures).
Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe (2005), Gradient Flows in Metric Spaces and in the Space of Probability Measures, ETH Zürich, Birkhäuser Verlag, Basel,
ISBN3-7643-2428-7
Athreya, Krishna B.; Lahiri, Soumendra N. (2006), Measure theory and probability theory, Springer,
ISBN0-387-32903-X
Bruckner, A. M.; Bruckner, J. B.; Thomson, B. S. (1997), Real Analysis, Prentice Hall,
ISBN0-134-58886-X