Singular measure

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 Rn is called singular, if it is singular in 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.