# Singular measure

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In mathematics, two positive (or signed or complex) measures μ and ν defined on a measurable space (Ω, Σ) are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A. This is denoted by $\mu \perp \nu .$ 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 $\mathbb {R} ^{n}$ 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,

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