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 ; 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 . 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 :