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.

A Dirac measure is a measure δ_x on a set X (with any σ-algebra of subsets of X) defined for a given x ∈ X and any (measurable) set A ⊆ X by

where 1_A is the indicator function of A.

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. 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.

Let δ_x denote the Dirac measure centred on some fixed point x in some measurable space (X, Σ).