In mathematics, two positive (or signed or complex) measures μ {\displaystyle \mu } and ν {\displaystyle \nu } defined on a measurable space ( Ω , Σ ) {\displaystyle (\Omega ,\Sigma )} are called singular if there exist two disjoint measurable sets A , B ∈ Σ {\displaystyle A,B\in \Sigma } whose union is Ω {\displaystyle \Omega } such that μ {\displaystyle \mu } is zero on all measurable subsets of B {\displaystyle B} while ν {\displaystyle \nu } is zero on all measurable subsets of A . {\displaystyle A.} This is denoted by μ ⊥ ν . {\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.
Examples on Rn As a particular case, a measure defined on the Euclidean space R n {\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,
H ( x ) = d e f { 0 , x < 0 ; 1 , x ≥ 0 ; {\displaystyle H(x)\ {\stackrel {\mathrm {def} }{=}}{\begin{cases}0,&x<0;\\1,&x\geq 0;\end{cases}}}
has the Dirac delta distribution δ 0 {\displaystyle \delta _{0}} as its distributional derivative. This is a measure on the real line, a "point mass" at 0. {\displaystyle 0.} However, the Dirac measure δ 0 {\displaystyle \delta _{0}} is not absolutely continuous with respect to Lebesgue measure λ , {\displaystyle \lambda ,} nor is λ {\displaystyle \lambda } absolutely continuous with respect to δ 0 : {\displaystyle \delta _{0}:} λ ( { 0 } ) = 0 {\displaystyle \lambda (\{0\})=0} but δ 0 ( { 0 } ) = 1 ; {\displaystyle \delta _{0}(\{0\})=1;} if U {\displaystyle U} is any non-empty open set not containing 0, then λ ( U ) > 0 {\displaystyle \lambda (U)>0} but δ 0 ( 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 R 2 . {\displaystyle \mathbb {R} ^{2}.}
The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
See also Absolute continuity (measure theory) – Form of continuity for functionsPages displaying short descriptions of redirect targets Lebesgue's decomposition theorem – Theorem in mathematical measure theory Singular distribution – Distribution concentrated on a set of measure zero
References
Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. ISBN 1-58488-347-2. J Taylor, An Introduction to Measure and Probability, Springer, 1996. ISBN 0-387-94830-9.
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