Dirac measure
# 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.

## 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

    $\delta _{x}(A)=1_{A}(x)={\begin{cases}0,&x\not \in A;\\1,&x\in A.\end{cases}}$

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

    $\int _{X}f(y)\,\mathrm {d} \delta _{x}(y)=f(x),$

which, in the form

    $\int _{X}f(y)\delta _{x}(y)\,\mathrm {d} y=f(x),$

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.
* The Dirac measure is a sigma-finite measure.

## 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.

