Dirac measure

A diagram showing all possible subsets of a 3-point set

{x, y, z}

. The Dirac measure

δ x

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.

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 $x \in X$ and any (measurable) set $A \subseteq X$ 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 ${\emptyset, X}$ .
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 $A = ℝn \ {x}$ 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

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.

Dirac measure

Definition

Properties of the Dirac measure

Generalizations

General references

See also