Rahul Sharma (Editor)

Singular measure

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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 μ ν .

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 )   = d e f { 0 , x < 0 ; 1 , x 0 ;

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

References

Singular measure Wikipedia