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\begin{document}

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\section*{Dirac measure}

[[!redirects Dirac measures]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_measurable_spaces}{For measurable spaces}\dotfill \pageref*{for_measurable_spaces} \linebreak
\noindent\hyperlink{for_topological_spaces}{For topological spaces}\dotfill \pageref*{for_topological_spaces} \linebreak
\noindent\hyperlink{for_locales}{For locales}\dotfill \pageref*{for_locales} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{significance}{Significance}\dotfill \pageref*{significance} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{for_measurable_spaces}{}\subsubsection*{{For measurable spaces}}\label{for_measurable_spaces}

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

\begin{displaymath}
\delta_x(A) \;\coloneqq\; 
\begin{cases}
1 & x\in A \\
0 & x\notin A
\end{cases}
\end{displaymath}
for each [[measurable set]] $A\subseteq X$.

\hypertarget{for_topological_spaces}{}\subsubsection*{{For topological spaces}}\label{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

\begin{displaymath}
\delta_x(U) \;\coloneqq\; 
\begin{cases}
1 & x\in U \\
0 & x\notin U
\end{cases}
\end{displaymath}
for each [[open set]] $U\subseteq X$.

Equivalently, it is the [[continuous valuation\#extending\_valuations\_to\_measures|extension to a measure]] of the [[continuous valuation\#dirac\_valuations|Dirac valuations]].

\hypertarget{for_locales}{}\subsubsection*{{For locales}}\label{for_locales}

(\ldots{})

(See also [[correspondence between measure and valuation theory]].)

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item On [[topological spaces]], Dirac measures are [[Radon measure|Radon]] and [[τ-additive]].


\item Every [[continuous valuation\#dirac\_valuations|Dirac valuation]] on a topological space can be [[continuous valuation\#extending\_valuations\_to\_measures|extended]] to a Dirac measure.


\item 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]] $\{x\}$.


\item The [[pushforward measure]] of a Dirac measure along a [[measurable function]] is again a Dirac measure. This is related to [[natural transformation|naturality]] of the unit map of [[probability and measure monads]].


\item 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 $\mu$.


\item The coupling above defines a map $X\times P Y\to P(X\times Y)$ which gives the [[strong monad|strength]] of most [[probability and measure monads]].



\end{itemize}
\hypertarget{significance}{}\subsection*{{Significance}}\label{significance}

\begin{itemize}%
\item The Dirac measures (and the [[continuous valuation\#Dirac valuations|Dirac valuations]]) give the unit of all [[probability and measure monads]].


\item The probabilistic interpretation is that the Dirac measures are exactly those of [[deterministic]] elements (or almost deterministic), i.e. which are ``not truly random''.


\item 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 $x$.



\end{itemize}
\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[Dirac distribution]]


\item [[monads of probability, measures, and valuations]]


\item [[measure]], [[Radon measure]], [[τ-additive measure]], [[continuous valuation]]



\end{itemize}
[[!redirects Dirac delta measure]] [[!redirects Dirac delta measures]] [[!redirects delta measure]] [[!redirects delta measures]] [[!redirects point measure]] [[!redirects point measures]] [[!redirects point mass measure]] [[!redirects point mass measures]]



\end{document}