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 for some m in the Lp space Lp(I). Properties of these generalizations Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous. 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 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 This 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: 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 for all Borel subsets of the real line. 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. Generalizations If and are two measures on the same measurable space is said to be with respect to if for every set for which This is written as "". That is: When then is said to be Absolute continuity of measures is reflexive and transitive,