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
[edit]A Dirac measure is a measure
where 1A 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; 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 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.
Properties of the Dirac measure
[edit]Let
δ x is a probability measure, and hence a finite measure.
Suppose that (X, T) is a topological space and that
δ x is a strictly positive measure if and only if the topology T is such that x lies within every non-empty open set, e.g. in the case of the trivial topology {∅, X}.- Since
δ x is probability measure, it is also a locally finite measure. - If X is a Hausdorff topological space with its Borel
σ -algebra, thenδ x satisfies the condition to be an inner regular measure, since singleton sets such as {x} are always compact. Hence,δ x is also a Radon measure. - Assuming that the topology T is fine enough that {x} is closed, which is the case in most applications, the support of
δ x is {x}. (Otherwise, supp(δ x) is the closure of {x} in (X, T).) Furthermore,δ x is the only probability measure whose support is {x}. - If X is n-dimensional Euclidean space Rn with its usual
σ -algebra and n-dimensional Lebesgue measureλ n, thenδ x is a singular measure with respect toλ n: simply decompose Rn as A = Rn \ {x} and B = {x} and observe thatδ x(A) =λ n(B) = 0. - The Dirac measure is a sigma-finite measure.
Generalizations
[edit]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
[edit]References
[edit]- Dieudonné, Jean (1976). "Examples of measures". Treatise on analysis, Part 2. Academic Press. p. 100. ISBN 0-12-215502-5.
- Benedetto, John (1997). "§2.1.3 Definition,
δ ". Harmonic analysis and applications. CRC Press. p. 72. ISBN 0-8493-7879-6.