right|thumb|250px|A diagram showing all possible subsets of a 3-point set }. The Dirac measure assigns a size of 1 to all sets in the upper-left half of the diagram and 0 to all sets in the lower-right half.

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 (with any -algebra of subsets of ) defined for a given and any (measurable) set by

:<math>\delta_x (A) = 1_A(x)= \begin{cases} 0, & x \not \in A; \\ 1, & x \in A. \end{cases}</math>

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 . 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

:<math>\int_{X} f(y) \, \mathrm{d} \delta_x (y) = f(x),</math>

which, in the form

:<math>\int_X f(y) \delta_x (y) \, \mathrm{d} y = f(x),</math>

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 .

  • is a probability measure, and hence a finite measure.

Suppose that is a topological space and that is at least as fine as the Borel -algebra on .

  • is a strictly positive measure if and only if the topology is such that lies within every non-empty open set, e.g. in the case of the trivial topology .
  • Since is probability measure, it is also a locally finite measure.
  • If is a Hausdorff topological space with its Borel -algebra, then satisfies the condition to be an inner regular measure, since singleton sets such as are always compact. Hence, is also a Radon measure.
  • Assuming that the topology is fine enough that is closed, which is the case in most applications, the support of is . (Otherwise, is the closure of in .) Furthermore, is the only probability measure whose support is .
  • If is -dimensional Euclidean space with its usual -algebra and -dimensional Lebesgue measure , then is a singular measure with respect to : simply decompose as and and observe that .
  • The Dirac measure is a sigma-finite measure.

Generalizations

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 (in respect to the Lebesgue measure) if its support is at most a countable set.

See also

  • Discrete measure
  • Dirac delta function

References