In mathematics, a Dirac measure is a measure δx on a set X (with any σ-algebra of subsets of X) defined for a given and any (measurable) set A ⊆ X 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 x in the sample space X. 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
which, in the form
is often taken to be part of the definition of the "delta function", holds as a theorem of Lebesgue integration.
Read more about Dirac Measure: Properties of The Dirac Measure, General References
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