Dirac measure

A diagram showing all possible subsets of a 3-point set Template:Math}. The Dirac measure Template:Math 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 Template:Math on a set Template:Math (with any [[sigma algebra|Template:Math-algebra]] of subsets of Template:Math) defined for a given Template:Math and any (measurable) set Template:Math by

${\delta }_x(A)=1_A(x)=\{\begin{array}{cc}0,& x\in ̸A;\\ 1,& x\in A.\end{array}$

where Template:Math is the indicator function of Template:Math.

The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome Template:Math in the sample space Template:Math. We can also say that the measure is a single atom at Template:Math; 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 sequenceTemplate:Dubious. The Dirac measures are the extreme points of the convex set of probability measures on Template:Math.

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

$\underset{X}{\int }f(y)\mathrm{d}{\delta }_x(y)=f(x),$

which, in the form

$\underset{X}{\int }f(y){\delta }_x(y)\mathrm{d}y=f(x),$

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

Let Template:Math denote the Dirac measure centred on some fixed point Template:Math in some measurable space Template:Math.

• Template:Math is a probability measure, and hence a finite measure.

Suppose that Template:Math is a topological space and that Template:Math is at least as fine as the [[Borel sigma algebra|Borel Template:Math-algebra]] Template:Math on Template:Math.

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

• Template:Cite book
• Template:Cite book