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
Examples on Rn
As a particular case, a measure defined on the Euclidean space 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.
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.
- 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.