Singular measure
# Singular measure

In mathematics, two positive (or signed or complex) measures $\mu$ and $\nu$ defined on a measurable space $(\Omega ,\Sigma )$ are called singular if there exist two disjoint measurable sets $A,B\in \Sigma$ 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 $\mu \perp \nu .$

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 $R^n$

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.

Example. A singular continuous measure on $\mathbb {R} ^{2}.$

The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.

