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 on a set defined for a given and any (measurable) set by
where is the indicator function of.
The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome in the sample space. We can also say that the measure is a single atom at. The Dirac measures are the extreme points of the convex set of probability measures on.
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
which, in the form
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 denote the Dirac measure centred on some fixed point in some measurable space.

is a probability measure, and hence a finite measure.

Suppose that is a topological space and that is at least as fine as the Borel -algebra on.

is a strictly positive measure if and only if the topology is such that lies within every non-empty open set, e.g. in the case of the trivial topology.
Since is probability measure, it is also a locally finite measure.
If is a Hausdorff topological space with its Borel -algebra, then satisfies the condition to be an inner regular measure, since singleton sets such as are always compact. Hence, is also a Radon measure.
Assuming that the topology is fine enough that is closed, which is the case in most applications, the support of is. Furthermore, is the only probability measure whose support is.
If is -dimensional Euclidean space with its usual -algebra and -dimensional Lebesgue measure, then is a singular measure with respect to : simply decompose as and and observe that.
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 if its support is at most a countable set.