HOME

TheInfoList



Click Here for Items Related To - singular measure
OR:
singular measure on:   [Wikipedia]   [Google]   [Amazon]

In 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 ...
is \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 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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
\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 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 ...
. 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 argum ...
on the real line, 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; if 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 su ...
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 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 ope ...
, 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.


See also

* * *


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)