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 (with any -algebra of subsets of ) 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 . (Otherwise, is the closure of in .) 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 (in respect to the Lebesgue measure) if its support is at most a countable set.

See also

• Discrete measure
• Dirac delta function

References