Dirac measure

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In mathematics, a Dirac measure is a measure $δ x$ on a set $X$ (with any sigma algebra of subsets of $X$ ) that gives the singleton set ${x}$ the measure 1, for a chosen element $x \in X$ :

$\delta_{x} \left( \{ x \} \right) = 1.$

In general, the measure is defined by

$\delta_{x} (A) = \left\{ \begin{matrix} 0, & x \not \in A; \\ 1, & x \in A. \end{matrix} \right.$

for any measurable set $A \subseteq X$ .

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 $x$ . (But 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

$\int_{X} f(y) \, \mathrm{d} \delta_{x} (y) = f(x),$