Dirac measure

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A diagram showing all possible subsets of a 3-point set \{x,y,z\}. The Dirac measure \delta_x 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 point x or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and engineering.

Definition[edit]

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

\delta_{x} (A) = 1_A(x)= \begin{cases} 0, & x \not \in A; \\ 1, & x \in A. \end{cases}

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; 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 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

\int_{X} f(y) \, \mathrm{d} \delta_{x} (y) = f(x),

which, in the form

\int_{X} f(y) \delta_{x} (y) \, \mathrm{d} y = f(x),

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

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

Suppose that (XT) is a topological space and that Σ is at least as fine as the Borel σ-algebra σ(T) on X.

  • δx 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 δx is probability measure, it is also a locally finite measure.
  • If X is a Hausdorff topological space with its Borel σ-algebra, then δx satisfies the condition to be an inner regular measure, since singleton sets such as {x} are always compact. Hence, δx 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 δx is {x}. (Otherwise, supp(δx) is the closure of {x} in (XT).) Furthermore, δx 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 measure λn, then δx is a singular measure with respect to λn: simply decompose Rn as A = Rn \ {x} and B = {x} and observe that δx(A) = λn(B) = 0.

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.

General references[edit]

See also[edit]