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 ${\displaystyle \mu \perp \nu .}$

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.

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,

has the Dirac delta distribution ${\displaystyle \delta _{0}}$ as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure ${\displaystyle \delta _{0}}$ is not absolutely continuous with respect to Lebesgue measure ${\displaystyle \lambda }$, nor is ${\displaystyle \lambda }$ absolutely continuous with respect to ${\displaystyle \delta _{0}}$: ${\displaystyle \lambda (\{0\})=0}$ but ${\displaystyle \delta _{0}(\{0\})=1}$; if ${\displaystyle U}$ is any open set not containing 0, then ${\displaystyle \lambda (U)>0}$ but ${\displaystyle \delta _{0}(U)=0}$.

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