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.

Definition

A Dirac measure is a measure on a set defined for a given and any set 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 in the sample space. We can also say that the measure is a single atom at ; 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.
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.

Properties of the Dirac measure

Let denote the Dirac measure centred on some fixed point in some measurable space.
Suppose that is a topological space and that is at least as fine as the Borel -algebra on.
A discrete measure is similar to the Dirac measure, except that it is concentrated at countably many points instead of a single point. More formally, a measure on the real line is called a discrete measure if its support is at most a countable set.

General references