Dirac measure

File:Hasse diagram of powerset of 3.svg

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 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

${\delta }_x(A)=1_A(x)=\{\begin{array}{ll}0,& x\in ̸A;\\ 1,& x\in A.\end{array}$

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. 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

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

which, in the form

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

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 Rⁿ with its usual σ-algebra and n-dimensional Lebesgue measure λⁿ, then δ_x is a singular measure with respect to λⁿ: simply decompose Rⁿ as A = Rⁿ \ {x} and B = {x} and observe that δ_x(A) = λⁿ(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.

See also

• Discrete measure
• Dirac delta function