# nLab Dirac measure

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## 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.

## Definition

### For measurable spaces

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 $A\subseteq X$.

### For topological spaces

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 $U\subseteq X$.

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

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## Properties

• 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.

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