mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, two positive (or signed or

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

) measures

\mu

and

\nu

defined on a

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...

(\Omega, \Sigma)

are called singular if there exist two disjoint measurable sets

A, B \in \Sigma

whose

union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...

\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 In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures \mu and \nu on a measurable space (\Omega,\Sigma), there exist two σ-finite signed measures \nu_0 and \nu_1 s ...

decomposes a singular measure into a singular continuous measure and a

discrete measure In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometric ...

. See below for examples.

Examples on R^''n''

As a particular case, a measure defined on the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

\R^n

is called ''singular'', if it is singular with respect to the

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

on this space. For example, the

Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

is a singular measure. Example. A

. The

Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...

on the

real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...

H(x) \ \stackrel \begin 0, & x < 0; \\ 1, & x \geq 0; \end

has the

Dirac delta distribution In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

\delta_0

as its

distributional derivative Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...

. This is a measure on the real line, a "

point mass A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...

" at

0.

However, the

Dirac measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. ...

\delta_0

is not absolutely continuous with respect to Lebesgue measure

\lambda,

nor is

\lambda

absolutely continuous with respect to

\delta_0:

\lambda(\) = 0

but

\delta_0(\) = 1;

U

is any

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...

not containing 0, then

\lambda(U) > 0

but

\delta_0(U) = 0.

Example. A singular continuous measure. The

Cantor distribution The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution has neither a probability density function nor a probability mass function, since although its cumulativ ...

has a

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...

that is continuous but not

absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...

, and indeed its absolutely continuous part is zero: it is singular continuous. Example. A singular continuous measure on

\R^2.

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

References

* Eric W Weisstein, ''CRC Concise Encyclopedia of Mathematics'', CRC Press, 2002. . * J Taylor, ''An Introduction to Measure and Probability'', Springer, 1996. . {{PlanetMath attribution, id=4002, title=singular measure Integral calculus Measures (measure theory)

Examples on R''n''

See also

References

Examples on R^''n''