Dirac measure

A diagram showing all possible subsets of a 3-point set $\{x,y,z\}$. The Dirac measure $\delta_x$ 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 ∈ 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 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.

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

• Jean Dieudonné (1976). "Examples of measures". Treatise on analysis, Part 2. Academic Press. p. 100. ISBN 0-12-215502-5.
• John Benedetto (1997). "§2.1.3 Definition, δ". Harmonic analysis and applications. CRC Press. p. 72. ISBN 0-8493-7879-6.

See also

• Discrete measure
• Dirac delta function