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
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