# Dirac measure

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

## Definition

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 is the indicator function of .

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 measure

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.

## Generalizations

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 references

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