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 ${x,y,z}$ on a set $δ_{x}$ (with any $δ_{x}$-algebra of subsets of $X$) defined for a given $σ$ and any (measurable) set $X$ by
 * $$\delta_x (A) = 1_A(x)= \begin{cases} 0, & x \not \in A; \\ 1, & x \in A. \end{cases}$$

where $x ∈ X$ is the indicator function of $A ⊆ X$.

The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome $1_{A}$ in the sample space $A$. 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 $x$ denote the Dirac measure centred on some fixed point $X$ in some measurable space $δ_{x}$.
 * $x$ is a probability measure, and hence a finite measure.

Suppose that $(X, Σ)$ is a topological space and that $δ_{x}$ is at least as fine as the Borel $(X, T)$-algebra $Σ$ on $σ$.
 * $σ(T)$ is a strictly positive measure if and only if the topology $X$ is such that $δ_{x}$ lies within every non-empty open set, e.g. in the case of the trivial topology $T$.
 * Since $x$ is probability measure, it is also a locally finite measure.
 * If ${∅, X}$ is a Hausdorff topological space with its Borel $δ_{x}$-algebra, then $X$ satisfies the condition to be an inner regular measure, since singleton sets such as $σ$ are always compact. Hence, $δ_{x}$ is also a Radon measure.
 * Assuming that the topology ${x}$ is fine enough that $δ_{x}$ is closed, which is the case in most applications, the support of $T$ is ${x}$. (Otherwise, $δ_{x}$ is the closure of ${x}$ in $supp(δ_{x})$.) Furthermore, ${x}$ is the only probability measure whose support is $(X, T)$.
 * If $δ_{x}$ is ${x}$-dimensional Euclidean space $X$ with its usual $n$-algebra and $R^{n}$-dimensional Lebesgue measure $σ$, then $n$ is a singular measure with respect to $λ^{n}$: simply decompose $δ_{x}$ as $λ^{n}$ and $R^{n}$ and observe that $A = R^{n} \ {x}$.
 * 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.