Singular measure

In mathematics, two positive (or signed or complex) measures $\mu$ and $\nu$ defined on a measurable space $(\mathrm{\Omega },\mathrm{\Sigma })$ are called singular if there exist two disjoint measurable sets $A,B\in \mathrm{\Sigma }$ whose union is $\mathrm{\Omega }$ 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 ${\mathbb{ℝ}}^n$ 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, $$H(x)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\{\begin{array}{ll}0,& x<0;\\ 1,& x\ge 0;\end{array}$$ 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 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 ${\mathbb{ℝ}}^2.$

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

See also

• Lebesgue's decomposition theorem – Theorem
• Singular distribution

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.