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In mathematics, one may talk about absolute continuity of functions and absolutely continuity of measures, and these two notions are closely connected.

Absolute continuity of functionsEdit


Let (X, d) be a metric space and let I be an interval in the real line R. A function f : IX is absolutely continuous on I if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint sub-intervals [xk, yk] of I, k = 1, 2, ..., n satisfies

\sum_{k=1}^{n} \left| y_k - x_k \right| < \delta


\sum_{k=1}^{n} d \left( f(y_k), f(x_k) \right) < \varepsilon.

The collection of all absolutely continuous functions from I into X is denoted AC(I; X).

A further generalisation is the space ACp(I; X) of curves f : IX such that

d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I

for some m in the Lp space Lp(I; R).


  • The Cantor function is continuous everywhere but not absolutely continuous; as is the function
f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x \sin(1/x), & \mbox{if } x \neq 0 \end{cases}
on a finite interval containing the origin, or the function f(x)=x^2 on an infinite interval.
  • If f : [a,b] → X is absolutely continuous, then it is of bounded variation on [a,b].
  • If f : [a,b] → R is absolutely continuous, then it has the Luzin N property (that is, for any L \subseteq [a,b] that \lambda(L)=0, it holds that \lambda(f(L))=0, where \lambda stands for the Lebesgue measure on R).
  • If f : IR is absolutely continuous, then f has a derivative almost everywhere.
  • If f : IR is continuous, is of bounded variation and has the Luzin N property, then it is absolutely continuous.
  • 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 mLp(I; R) such that
d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I.

Absolute continuity of measuresEdit

If μ and ν are measures on the same measure space (or, more precisely, on the same sigma-algebra) then μ is absolutely continuous with respect to ν if μ(A) = 0 for every set A for which ν(A) = 0. It is written as "μ << ν". In symbols:

\mu \ll \nu \iff \left( \nu(A) = 0 \implies \mu (A) = 0 \right).

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 states that if μ is absolutely continuous with respect to ν, and ν is σ-finite, then μ has a density, or "Radon-Nikodym derivative", with respect to ν, which implies 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

\mu(A)=\int_A f\,d\nu.

The connection between absolute continuity of real functions and absolute continuity of measuresEdit

A 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.

Example. The Heaviside step function on the real line,

H(x) \ \stackrel{\mathrm{def}}{=} \ \left\{ \begin{matrix} 0, & x < 0; \\ 1, & x \geq 0; \end{matrix} \right.

has the Dirac delta distribution \delta_{0} as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure \delta_{0} is not absolutely continuous with respect to Lebesgue measure \lambda, nor is \lambda absolutely continuous with respect to \delta_{0}: \lambda ( \{ 0 \} ) = 0 but \delta_{0} ( \{ 0 \} ) = 1; if U is any open set not containing 0, then \lambda (U) > 0 but \delta_{0} (U) = 0.

Example. The Cantor distribution has a continuous cumulative distribution function, but nonetheless the Cantor distribution is not absolutely continuous with respect to Lebesgue measure.

See alsoEdit


  • Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures, ETH Zürich, Birkhäuser Verlag, Basel. ISBN 3-7643-2428-7.
  • Royden, H.L. (1968). Real Analysis, Collier Macmillan. ISBN 0-02-979410-2.
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