# 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.

## Contents

## DefinitionEdit

A **Dirac measure** is a measure *δ*_{x} on a set *X* (with any *σ*-algebra of subsets of *X*) defined for a given *x* ∈ *X* and any (measurable) set *A* ⊆ *X* by

where 1_{A} is the indicator function of *A*.

The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome *x* in the sample space *X*. We can also say that the measure is a single atom at *x*; 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. The Dirac measures are the extreme points of the convex set of probability measures on *X*.

The name is a back-formation from the Dirac delta function, considered as a Schwartz distribution, for example on the real line; measures can be taken to be a special kind of distribution. The identity

which, in the form

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

## Properties of the Dirac measureEdit

Let *δ*_{x} denote the Dirac measure centred on some fixed point *x* in some measurable space (*X*, *Σ*).

*δ*_{x}is a probability measure, and hence a finite measure.

Suppose that (*X*, *T*) is a topological space and that *Σ* is at least as fine as the Borel *σ*-algebra *σ*(*T*) on *X*.

*δ*_{x}is a strictly positive measure if and only if the topology*T*is such that*x*lies within every non-empty open set, e.g. in the case of the trivial topology {∅,*X*}.- Since
*δ*_{x}is probability measure, it is also a locally finite measure. - If
*X*is a Hausdorff topological space with its Borel*σ*-algebra, then*δ*_{x}satisfies the condition to be an inner regular measure, since singleton sets such as {*x*} are always compact. Hence,*δ*_{x}is also a Radon measure. - Assuming that the topology
*T*is fine enough that {*x*} is closed, which is the case in most applications, the support of*δ*_{x}is {*x*}. (Otherwise, supp(*δ*_{x}) is the closure of {*x*} in (*X*,*T*).) Furthermore,*δ*_{x}is the only probability measure whose support is {*x*}. - If
*X*is*n*-dimensional Euclidean space**ℝ**^{n}with its usual*σ*-algebra and*n*-dimensional Lebesgue measure*λ*^{n}, then*δ*_{x}is a singular measure with respect to*λ*^{n}: simply decompose**ℝ**^{n}as*A*=**ℝ**^{n}\ {*x*} and*B*= {*x*} and observe that*δ*_{x}(*A*) =*λ*^{n}(*B*) = 0. - The Dirac measure is a sigma-finite measure

## GeneralizationsEdit

A discrete measure is similar to the Dirac measure, except that it is concentrated at countably many points instead of a single point. More formally, a measure on the real line is called a **discrete measure** (in respect to the Lebesgue measure) if its support is at most a countable set.

## General referencesEdit

- Dieudonné, Jean (1976). "Examples of measures".
*Treatise on analysis, Part 2*. Academic Press. p. 100. ISBN 0-12-215502-5. - Benedetto, John (1997). "§2.1.3 Definition,
*δ*".*Harmonic analysis and applications*. CRC Press. p. 72. ISBN 0-8493-7879-6.