Dirac measure

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.

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[dubious – discuss].

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.