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 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, $$H(x) \ \stackrel{\mathrm{def}}{=} \begin{cases} 0, & x < 0; \\ 1, & x \geq 0; \end{cases}$$ 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 non-empty 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 $$\R^2.$$

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