, two positive (or signed
) measures μ
defined on a measurable space
(Ω, Σ) are called singular
if there exist two disjoint sets A
in Σ whose union
is Ω such that μ
is zero on all measurable subsets of B
is zero on all measurable subsets of A
. This is denoted by
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,
has the Dirac delta distribution as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure is not absolutely continuous with respect to Lebesgue measure , nor is absolutely continuous with respect to : but ; if is any open set not containing 0, then but .
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.
- 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.