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\section*{Radon monad}

\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{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{spaces_of_radon_measures}{Spaces of Radon measures}\dotfill \pageref*{spaces_of_radon_measures} \linebreak
\noindent\hyperlink{in_terms_of_additive_measures}{In terms of τ-additive measures}\dotfill \pageref*{in_terms_of_additive_measures} \linebreak
\noindent\hyperlink{in_terms_of_valuations}{In terms of valuations}\dotfill \pageref*{in_terms_of_valuations} \linebreak
\noindent\hyperlink{functoriality_unit_and_multiplication}{Functoriality, unit and multiplication}\dotfill \pageref*{functoriality_unit_and_multiplication} \linebreak
\noindent\hyperlink{algebras}{Algebras}\dotfill \pageref*{algebras} \linebreak
\noindent\hyperlink{the_ordered_case}{The ordered case}\dotfill \pageref*{the_ordered_case} \linebreak
\noindent\hyperlink{construction_of_the_spaces}{Construction of the spaces}\dotfill \pageref*{construction_of_the_spaces} \linebreak
\noindent\hyperlink{algebras_2}{Algebras}\dotfill \pageref*{algebras_2} \linebreak
\noindent\hyperlink{lax_morphisms_are_concave_maps}{Lax morphisms are concave maps}\dotfill \pageref*{lax_morphisms_are_concave_maps} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The Radon monad is a [[measure monad]] on the [[category]] of [[compact Hausdorff spaces]]. Its functor assigns to each space $X$ the space of [[Radon measure|Radon]] [[probability measure|probability]] or subprobability [[measures]], equipped with the [[weak topology]].

There is also an [[order theory|ordered]] version of the Radon monad on the category of [[compact ordered spaces]] (see \hyperlink{the_ordered_case}{below}).

\hypertarget{spaces_of_radon_measures}{}\subsection*{{Spaces of Radon measures}}\label{spaces_of_radon_measures}

Let $X$ be a [[compact Hausdorff space]]. Denote by $R X$ the set of those [[Radon measures]] on $X$ which are \emph{subnormalized}, i.e. those measures $\mu$ for which $\mu(X)\le 1$. Denote by $P X$ its subset of Radon [[probability measures]].

Equip both sets with the [[weak topology|topology of weak convergence]] with respect to [[continuous functions]]. It is well known that the resulting spaces $R X$ and $P X$ are themselves [[compact Hausdorff spaces]].

\hypertarget{in_terms_of_additive_measures}{}\subsubsection*{{In terms of τ-additive measures}}\label{in_terms_of_additive_measures}

Since [[tau-additive measure\#relationship\_with\_other\_measuretheoretical\_notions|on compact Hausdorff spaces Radon and τ-additive measures coincide]], we can equivalently define $R X$ and $P X$ as the sets of subnormalized and normalized [[τ-additive measures]].

For probability measures, the [[weak topology]] and the [[extended probabilistic powerdomain\#the\_measure\_monad\_on\_top|A-topology]] coincide, so that $P$ an be extended to the [[extended probabilistic powerdomain\#the\_probability\_monad\_on\_top|probability monad on Top]].

The same cannot be said for $R$, since the weak topology and the A-topology do not coincide for \emph{subnormalized} measures: given $\mu\in R X$, the measure $r \mu$ for $0 \le r \le 1$ is in the [[closure]] of $\mu$ for the $A$-topology, while the weak topology is [[Hausdorff]].

\hypertarget{in_terms_of_valuations}{}\subsubsection*{{In terms of valuations}}\label{in_terms_of_valuations}

Note that, since every [[continuous valuation]] on a [[compact Hausdorff space]] can be [[continuous valuation\#extending\_valuations\_to\_measures|extended to a τ-additive measure]], we could equivalently defined the Radon monad as a monad of [[continuous valuations]].

Just as above, for probability measures, the [[weak topology]] and the [[extended probabilistic powerdomain\#spaces\_of\_valuations|pointwise topology of valuations]] coincide, so that $P$ an be extended to a submonad of the [[extended probabilistic powerdomain]] on [[Top]]. Again, the same cannot be said for $R$.

\hypertarget{functoriality_unit_and_multiplication}{}\subsection*{{Functoriality, unit and multiplication}}\label{functoriality_unit_and_multiplication}

Let $f:X\to Y$ be a [[continuous map]] between [[compact Hausdorff spaces]]. The [[pushforward of measures]] (or equivalently of [[continuous valuation\#pushforward|valuations]]) gives a well-defined, continuous map $R X\to R Y$ which restricts to the map $P X \to P Y$. This makes $R$ and $P$ [[endofunctors]] of [[Top]].

As it is usual for [[measure monads]], the unit is given by the [[Dirac measures]] and the multiplication is given by [[integration]].

More in detail, given a [[compact Hausdorff space]] $X$, we can assign to each $x\in X$ its [[Dirac measure]] (equivalently, [[continuous valuation\#dirac\_valuations|Dirac valuation]] $\delta_x$. The assignment $\delta:X\to P X$, or $X\to R X$ is continuous, and [[natural transformation|natural]] in $X$.

Just as well, given a measure $\mu\in P P X$, we can define the measure $E\mu\in P X$ as the one assigning to a measurable $A\subseteq X$ the number

\begin{displaymath}
E\mu(A) \coloneqq \int_{P X} p(A) \,d\mu(p).
\end{displaymath}
(In terms of valuations, the same is given for [[open sets]] instead of [[measurable sets]].) Again, the assignment $E: P P X \to P X$ is continuous and [[natural transformation|natural]] in $X$. The multiplication for $R$ is defined analogously.

The unit and multiplication thus defined satisfy the usual [[axioms]] of a [[monad]]. The monads $R$ and $P$ are both known in the literature as the \textbf{Radon monad}.

The monad $P$ is the restriction to [[compact Hausdorff spaces]] of the [[extended probabilistic powerdomain\#the\_probability\_monad\_on\_top|probability monad on Top]], which is itself a submonad of the [[extended probabilistic powerdomain]]. (See also [[monads of probability, measures, and valuations]].)

\hypertarget{algebras}{}\subsection*{{Algebras}}\label{algebras}

The [[algebra over a monad|algebras]] of the Radon monad $P$ are [[compact]] [[convex subsets]] of [[locally convex topological vector spaces]].

More in detail, a compact convex subset $C$ of a locally convex topological vector space is a [[compact Hausdorff space]], and it admits a canonical $P$-algebra structure $e:P C\to C$ via (vector-valued) [[integration]]:

\begin{displaymath}
p \mapsto \int_{C} c \,dp(c) .
\end{displaymath}
Since the space is [[compact]], the integral above is well-defined, and it returns an element of $C$ which we can view as the ``center of mass'' of $p$.

Conversely, it can be proven that every $P$-algebra is of this form.

The [[Eilenberg-Moore category|morphisms of algebras]] are the continuous maps between algebras which commute with the operation of taking integrals. It can be shown that these coincide with the \emph{affine} maps, i.e. those maps $f:A\to B$ which satisfy

\begin{displaymath}
f\big( \lambda\, a + \mu\, b  \big) \;=\; \lambda\,f(a) + \mu\,f(b)
\end{displaymath}
for all $a,b\in A$ and $0\le\lambda,\mu\le 1$ (with $\lambda+\mu=1$ for the normalized case).

For more information, see \hyperlink{swirszcz}{Swirszcz `74} and the later \hyperlink{radonkeimel}{Keimel `08}.

\hypertarget{the_ordered_case}{}\subsection*{{The ordered case}}\label{the_ordered_case}

The Radon monad can be lifted to the [[CompOrd\#categories\_of\_compact\_ordered\_spaces|category of compact ordered spaces]] (and [[continuous map|continuous]] [[monotone maps]]). This is done by means of the [[stochastic order]].

Here we sketch the construction. For more details, see \hyperlink{radonkeimel}{Keimel `08}.

\hypertarget{construction_of_the_spaces}{}\subsubsection*{{Construction of the spaces}}\label{construction_of_the_spaces}

Given a [[compact ordered space]] $X$, we can form the spaces $R X$ and $P X$ as for the unordered case, and then equip them with the [[stochastic order]]. Over compact ordered spaces, the stochastic order is antisymmetric (i.e. it is a partial order), and it has [[closed graph]], so that $R X$ and $P X$ are again compact ordered spaces.

Given a monotone map $f:X\to Y$, the maps $R f$ and $P f$ are monotone for the stochastic order, and so are the monad structure maps. This way, $P$ and $R$ lift to a monad on [[CompOrd\#categories\_of\_compact\_ordered\_spaces|CompOrd]].

For the canonical [[locally posetal 2-category]] structure of [[CompOrd\#categories\_of\_compact\_ordered\_spaces|CompOrd]] given by the [[pointwise order]], $P$ and $R$ are even [[2-monads]], since whenever $f\le g:X\to Y$, we have $R f\le R g$ $P f\le P g$.

Both the resulting monads are known in the literature as \textbf{ordered Radon monad}.

\hypertarget{algebras_2}{}\subsubsection*{{Algebras}}\label{algebras_2}

The algebras of the ordered Radon monad $P$ are known to be [[compact]] [[convex space|convex]] subsets of [[locally convex topological vector space|locally convex]] [[ordered topological vector spaces]]. This can be seen as the ordered equivalent of the characterization \hyperlink{algebras}{above}.

The algebras of $R$, similarly, can be seen as the ordered equivalent of [[Kegelspitze]].

See \hyperlink{radonkeimel}{Keimel `08} for more.

Just as for the unordered case, the [[Eilenberg-Moore category|algebra morphisms]] are the continuous affine maps, which here are also required to be monotone.

\hypertarget{lax_morphisms_are_concave_maps}{}\subsubsection*{{Lax morphisms are concave maps}}\label{lax_morphisms_are_concave_maps}

Differently from the unordered case, in the ordered setting we have a [[2-category]], and so it makes sense to talk about [[lax morphisms]] of algebras. By definition, these amount to maps between algebras $f:A\to B$ with the [[stuff, structure, property|property]] that

\begin{displaymath}
f \left( \int a \, dp(a) \right) \;\le\; \int f(a) \, dp(a)
\end{displaymath}
for all $p\in P A$ (resp. $R A$).

By the generalized [[Jensen's inequality]], these are precisely the (continuous, monotone) [[concave maps]], i.e. the maps that satisfy

\begin{displaymath}
f\big( \lambda\, a + \mu\, b  \big) \;\le\; \lambda\,f(a) + \mu\,f(b) .
\end{displaymath}
(Compare with the strict case by replacing the order with equalities.)

Just as well, the oplax morphisms are the (continuous, monotone) [[convex maps]].

(For more information see the analogous discussion for the [[Kantorovich monad]] in \hyperlink{orderedkantorovich}{F-P `18}.)

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

\begin{itemize}%
\item [[monads of probability, measures, and valuations]]


\item [[Giry monad]], [[extended probabilistic powerdomain]], [[Kantorovich monad]]


\item [[locally convex topological vector space]], [[ordered vector space]], [[convex space]]


\item [[compact Hausdorff space]], [[compact ordered space]], [[stably compact space]]


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



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item T. Swirszcz, \emph{Monadic functors and convexity}, Bulletin de l'Academie Polonais des Sciences 22, 1974 (\href{https://www.fuw.edu.pl/~kostecki/scans/swirszcz1974.pdf}{pdf})


\item [[Klaus Keimel]], \emph{The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras}, Topology and its Applications, 2008 (\href{https://doi.org/10.1016/j.topol.2008.07.002}{doi:10.1016/j.topol.2008.07.002})


\item [[Tobias Fritz]] and Paolo Perrone, \emph{Stochastic order on metric spaces and the ordered Kantorovich monad}, submitted, 2018. (\href{https://arxiv.org/abs/1808.09898}{arXiv:1808.09898})



\end{itemize}
[[!redirects ordered Radon monad]] [[!redirects radon monad]] [[!redirects ordered radon monad]]



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