Dirac measure
In mathematics, a Dirac measure is a measure δx on a set X that gives a given element x measure 1, so that
- δx({x}) = 1
and in general
- δx(Y) = 0
for any subset Y of X not containing x,
- δx(Z) = 1
for any subset Z containing x.
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. 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.