nLab Dirac measure

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Idea

A Dirac measure is a measure whose (unit) mass is concentrated on a single point $x$ of a space $X$ .

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 $x$ .

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

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

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

for each measurable set $A\subseteq X$ .

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

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

for each open set $U\subseteq X$ .

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On topological spaces, Dirac measures are Radon and τ-additive.
Every Dirac valuation on a topological space can be extended to a Dirac measure.
On a topological space $X$ , the support of the Dirac measure at $x\in X$ is equal to the closure of $x$ . On T1 spaces, this is just the singleton $\{x\}$ .
The pushforward measure of a Dirac measure along a measurable function is again a Dirac measure. This is related to naturality of the unit map of probability and measure monads.
Given a Dirac measure $\delta_x$ on a measurable space $X$ and any measure $\mu$ on any measurable space $Y$ , the product measure? $\delta_x\otimes \mu$ is the unique coupling of $\delta_x$ and $\mu$ .
The coupling above defines a map $X\times P Y\to P(X\times Y)$ which gives the strength of most probability and measure monads.

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 $X$ has the Dirac measure $\delta_x$ as law if and only if it is almost surely equal to $x$ .