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

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 ${\displaystyle \mathbb {R} ^{n))$ 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,

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

${\displaystyle \delta _{0))$

${\displaystyle 0.}$

${\displaystyle \delta _{0))$

${\displaystyle \lambda ,}$

${\displaystyle \lambda }$

${\displaystyle \delta _{0}:}$

${\displaystyle \lambda (\{0\})=0}$

${\displaystyle \delta _{0}(\{0\})=1;}$

${\displaystyle U}$

${\displaystyle \lambda (U)>0}$

${\displaystyle \delta _{0}(U)=0.}$

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 ${\displaystyle \mathbb {R} ^{2}.}$

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