but is not antisymmetric, so it is a preorder rather than a partial order. Instead, if and the measures and are said to be equivalent. Thus absolute continuity induces a partial ordering of such equivalence classes. 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 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, 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. More generally, the measure μ is assumed to be locally finite (rather than finite) and F(x) is defined as μ((0,x]) for , 0 for , and −μ((x,0]) for . In this case μ is the Lebesgue–Stieltjes measure generated by F. The relation between the two notions of absolute continuity still holds. Notes References Leoni, Giovanni (2009), A First Course in Sobolev Spaces, Graduate Studies in Mathematics, American Mathematical Society, pp. xvi+607 , , , MAA External links Absolute continuity at Encyclopedia of Mathematics Topics in Real and Functional Analysis by Gerald Teschl Theory of continuous functions Real analysis Measure theory