nLab
Dirac measure

Contents

Contents

Idea

A Dirac measure is a measure whose (unit) mass is concentrated on a single point xx of a space XX.

From the point of view of probability theory, a Dirac measure can be seen as the law? of a deterministic random variable?, or more generally one which is almost surely? equal to a point xx.

See also Dirac distribution for the analogous concept in the language of distributions.

Definition

For measurable spaces

Let XX be a measurable space. Given xXx\in X, the Dirac measure δ x\delta_x at xx is the measure defined by

δ x(A){1 xA 0 xA \delta_x(A) \;\coloneqq\; \begin{cases} 1 & x\in A \\ 0 & x\notin A \end{cases}

for each measurable set AXA\subseteq X.

For topological spaces

If XX is a topological space, the Dirac measure at xx can be also defined as the unique Borel measure δ x\delta_x which satisfies

δ x(U){1 xU 0 xU \delta_x(U) \;\coloneqq\; \begin{cases} 1 & x\in U \\ 0 & x\notin U \end{cases}

for each open set UXU\subseteq X.

Equivalently, it is the extension to a measure of the Dirac valuations.

For locales

(…)

(See also correspondence between measure and valuation theory.)

Properties

Significance

  • The Dirac measures (and the Dirac valuations) give the unit of all probability and measure monads.

  • The probabilistic interpretation is that the Dirac measures are exactly those of deterministic? elements (or almost deterministic), i.e. which are “not truly random”.

  • In terms of random variables, and somewhat conversely, a random element? of XX has the Dirac measure δ x\delta_x as law? if and only if it is almost surely? equal to xx.

See also

Last revised on October 27, 2019 at 19:20:33. See the history of this page for a list of all contributions to it.