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 ...

, 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 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 ...

, an important tool in physics and other technical fields.

Definition

A Dirac measure is a

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

on a set (with any -algebra of

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of ) defined for a given and any (measurable) set by :

\delta_x (A) = 1_A(x)= \begin 0, & x \not \in A; \\ 1, & x \in A. \end

where is the

indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...

of . The Dirac measure is a

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...

, and in terms of probability it represents the

almost sure In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...

outcome in the

sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...

. We can also say that the measure is a single

atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, and ...

at ; however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a

delta sequence Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, at a river mouth * D (NATO phonetic alphabet: "Delta") * Delta Air Lines, US * Delta variant of SARS-CoV-2 that causes COVID-19 Delta may also re ...

. The Dirac measures are the

extreme point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or ...

s of the convex set of probability measures on . The name is a back-formation from the

; considered as a

Schwartz distribution 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 differentiate functions whose derivatives d ...

, for example 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 ...

, measures can be taken to be a special kind of distribution. The identity :

\int_ f(y) \, \mathrm \delta_x (y) = f(x),

which, in the form :

\int_X f(y) \delta_x (y) \, \mathrm y = f(x),

is often taken to be part of the definition of the "delta function", holds as a theorem of

Lebesgue integration In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...

Properties of the Dirac measure

Let denote the Dirac measure centred on some fixed point in some

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 ...

. * is a probability measure, and hence a

finite measure In measure theory, a branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than mor ...

. Suppose that is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

and that is at least as fine as the Borel -algebra on . * is a

strictly positive measure In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points". Definition Let (X, T) be a Hausdorff topological space and let \Sigma be ...

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

the topology is such that lies within every non-empty open set, e.g. in the case of the

trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...

. * Since is probability measure, it is also a

locally finite measure In mathematics, a locally finite measure is a Measure (mathematics), measure for which every point of the measure space has a Neighbourhood (mathematics), neighbourhood of Finite set, finite measure. Definition Let (X, T) be a Hausdorff space, Hau ...

. * If is a Hausdorff topological space with its Borel -algebra, then satisfies the condition to be an

inner regular measure In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. Definition Let (''X'', ''T'') be a Hausdorff topological space and let Σ be a σ-algebra on ''X'' tha ...

, since

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...

sets such as are always

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

. Hence, is also a

Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...

. * Assuming that the topology is fine enough that is closed, which is the case in most applications, the

support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...

of is . (Otherwise, is the closure of in .) Furthermore, is the only probability measure whose support is . * If is -dimensional

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 ...

with its usual -algebra and -dimensional

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 ...

, then is a

singular measure In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero on ...

with respect to : simply decompose as and and observe that . * The Dirac measure is a sigma-finite measure.

Generalizations

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 ...

is similar to the Dirac measure, except that it is concentrated at countably many points instead of a single point. More formally, a

on the

is called a discrete measure (in respect to the

) if its

is at most a

countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

References

* * {{DEFAULTSORT:Dirac Measure Measures (measure theory)

Definition

Properties of the Dirac measure

Generalizations

See also

References