# 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.

A diagram showing all possible subsets of a 3-point set {x,y,z}. The Dirac measure δx assigns a size of 1 to all sets in the upper-left half of the diagram and 0 to all sets in the lower-right half.

## Definition

A Dirac measure is a measure δx on a set X (with any σ-algebra of subsets of X) defined for a given xX and any (measurable) set AX by

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

where 1A 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[dubious ]. 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

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

which, in the form

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

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 δx denote the Dirac measure centred on some fixed point x in some measurable space (X, Σ).

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.

## 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.