In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space are called singular if there exist two disjoint measurable sets whose union is \Omega such that \mu is zero on all measurable subsets of B while \nu is zero on all measurable subsets of A. This is denoted by A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples.

Examples on Rn

As a particular case, a measure defined on the Euclidean space \R^n is called singular, if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure. Example. A discrete measure. The Heaviside step function on the real line, 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: but if U is any non-empty open set not containing 0, then but Example. A singular continuous measure. The Cantor distribution has a cumulative distribution function that is continuous but not absolutely continuous, and indeed its absolutely continuous part is zero: it is singular continuous. Example. A singular continuous measure on \R^2. The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.