Singular measure
In mathematics, two positive (or signed or complex) measures [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math] defined on a measurable space [math]\displaystyle{ (\Omega, \Sigma) }[/math] are called singular if there exist two disjoint measurable sets [math]\displaystyle{ A, B \in \Sigma }[/math] whose union is [math]\displaystyle{ \Omega }[/math] such that [math]\displaystyle{ \mu }[/math] is zero on all measurable subsets of [math]\displaystyle{ B }[/math] while [math]\displaystyle{ \nu }[/math] is zero on all measurable subsets of [math]\displaystyle{ A. }[/math] This is denoted by [math]\displaystyle{ \mu \perp \nu. }[/math] 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 [math]\displaystyle{ \R^n }[/math] 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, [math]\displaystyle{ H(x) \ \stackrel{\mathrm{def}}{=} \begin{cases} 0, & x \lt 0; \\ 1, & x \geq 0; \end{cases} }[/math] has the Dirac delta distribution [math]\displaystyle{ \delta_0 }[/math] as its distributional derivative. This is a measure on the real line, a "point mass" at [math]\displaystyle{ 0. }[/math] However, the Dirac measure [math]\displaystyle{ \delta_0 }[/math] is not absolutely continuous with respect to Lebesgue measure [math]\displaystyle{ \lambda, }[/math] nor is [math]\displaystyle{ \lambda }[/math] absolutely continuous with respect to [math]\displaystyle{ \delta_0: }[/math] [math]\displaystyle{ \lambda(\{0\}) = 0 }[/math] but [math]\displaystyle{ \delta_0(\{0\}) = 1; }[/math] if [math]\displaystyle{ U }[/math] is any open set not containing 0, then [math]\displaystyle{ \lambda(U) \gt 0 }[/math] but [math]\displaystyle{ \delta_0(U) = 0. }[/math]
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 [math]\displaystyle{ \R^2. }[/math]
The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
See also
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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