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\section*{extended probabilistic powerdomain}

[[!redirects Extended probabilistic powerdomain]]

\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{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{spaces_of_valuations}{Spaces of valuations}\dotfill \pageref*{spaces_of_valuations} \linebreak
\noindent\hyperlink{functoriality}{Functoriality}\dotfill \pageref*{functoriality} \linebreak
\noindent\hyperlink{unit_and_multiplication}{Unit and multiplication}\dotfill \pageref*{unit_and_multiplication} \linebreak
\noindent\hyperlink{monoidal_structure}{Monoidal structure}\dotfill \pageref*{monoidal_structure} \linebreak
\noindent\hyperlink{algebras}{Algebras}\dotfill \pageref*{algebras} \linebreak
\noindent\hyperlink{notable_submonads}{Notable submonads}\dotfill \pageref*{notable_submonads} \linebreak
\noindent\hyperlink{normalized_valuations}{Normalized valuations}\dotfill \pageref*{normalized_valuations} \linebreak
\noindent\hyperlink{the_measure_monad_on_top}{The measure monad on Top}\dotfill \pageref*{the_measure_monad_on_top} \linebreak
\noindent\hyperlink{the_probability_monad_on_top}{The probability monad on Top}\dotfill \pageref*{the_probability_monad_on_top} \linebreak
\noindent\hyperlink{the_monad_of_topological_cones}{The monad of topological cones}\dotfill \pageref*{the_monad_of_topological_cones} \linebreak
\noindent\hyperlink{the_monad_of_topological_convex_spaces}{The monad of topological convex spaces}\dotfill \pageref*{the_monad_of_topological_convex_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The extended probabilistic powerdomain is a [[monad of valuations]] on [[topological spaces]]. Its functor part assigns to a given [[topological space]] the space of [[continuous valuations]] over it.

The idea of this monad was first given by Kirch for the case of [[domains]] (see \hyperlink{kirch}{Kirch `93}, in German, or \hyperlink{Jung}{AJK `06}, in English). It was extended to all of [[Top]] and given its current form by Heckmann (see \hyperlink{Heckmann96}{Heckmann `96}).

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{spaces_of_valuations}{}\subsubsection*{{Spaces of valuations}}\label{spaces_of_valuations}

Given a [[topological space]] $X$, denote by $V X$ the space whose points are [[continuous valuations]] on $X$ with values in $[0,\infty]$. Equip $V X$ with the [[topology]] [[topological basis|generated]] by the sets in the form

\begin{displaymath}
\theta(U,r) \coloneqq \{ \nu \in V X \,|\, \nu(U) \gt r \},
\end{displaymath}
for $r\ge 0$ and $U\subseteq X$ [[open]], or equivalently by the sets in the form

\begin{displaymath}
\Theta(g,r) \coloneqq \left\{ \nu \in V X \,\Big|\, \int g \,d\nu \gt r \right\}
\end{displaymath}
for $r\ge 0$ and $g:X\to[0,\infty]$ [[lower semicontinuous]].

This topology can be seen as the [[pointwise topology]] if we view valuations either as functions on the [[open sets]] or as [[functionals]] on lower semicontinuous functions (via [[continuous valuation\#integration|integration]]). It is also the [[initial topology]] of either evaluation of open sets or integration of functions, meaning that it is the coarsest topology for which either the assignments

\begin{displaymath}
\nu \mapsto \nu(U)
\end{displaymath}
for every open $U\subseteq X$, or

\begin{displaymath}
\nu \mapsto \int g \, d\nu
\end{displaymath}
for every lower semicontinuous $g:X\to[0,\infty]$, are lower semicontinuous. Lower semicontinuity, in some sense, plays the role that [[measurable function|measurability]] plays for the [[Giry monad]] (see also [[correspondence between measure and valuation theory]]).

The [[specialization preorder]] of this topology is known as the [[stochastic order]], and can be seen as the [[pointwise order]] of valuations as functions on the open sets.

It is known (see \hyperlink{jung2}{Jung `04}) that if $X$ is [[stably compact]], then $V X$ is stably compact too, so that the monad $V$ restrict to the subcategory of [[stably compact spaces]].

\hypertarget{functoriality}{}\subsubsection*{{Functoriality}}\label{functoriality}

Given a [[continuous map]] $f:X\to Y$, define the map $f:V X\to V Y$ as the one assigning to a continuous valuation $\nu\in V X$ its [[continuous valuation\#pushforward|pushforward]] along $f$.

It can be proven that the map $V f$ is continuous too, so that $V$ is an [[endofunctor]] of [[Top]].

\hypertarget{unit_and_multiplication}{}\subsubsection*{{Unit and multiplication}}\label{unit_and_multiplication}

We can define the unit of the [[monad]] as follows. Given a space $X$, define the map $\delta:X\to V X$ as the one assigning to the point $x\in X$ the [[valuation (measure theory)\#dirac\_valuation|Dirac valuation]] at $x$. This map $\delta$ is [[continuous map|continuous]], and [[natural transformation|natural]] in $X$.

The multiplication map makes use of the concept of [[valuation (measure theory)\#integration|integration over a valuation]]. Given a valuation $\xi\in V V X$, we can define the valuation $E \xi\in V X$ as the one mapping an [[open]] set $U\subseteq X$ to

\begin{displaymath}
E \xi (U) \coloneqq \int_X \nu(U) \, d\xi(\nu).
\end{displaymath}
This integral is well defined, since the assignment $U\mapsto \nu(U)$ is [[lower semicontinuous]]. The assignment $U\mapsto E \xi (U)$ gives a [[continuous valuation]] on $X$, and the resulting map $E: V V X \to V X$ is [[continuous map|continuous]] and [[natural transformation|natural]] in $X$.

The maps $\delta$ and $E$ satisfy the usual [[axioms]] of a [[monad]]. The monad $(V,\delta,E)$ is usually called the \textbf{extended probabilistic powerdomain}.

This construction, especially the way the unit and multiplications are defined, can be thought of as a topological analogue of the [[Giry monad]].

\hypertarget{monoidal_structure}{}\subsection*{{Monoidal structure}}\label{monoidal_structure}

(\ldots{})

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

(\ldots{})

\hypertarget{notable_submonads}{}\subsection*{{Notable submonads}}\label{notable_submonads}

There are a number of monads that can be constructed as submonads of $V$. The [[monoidal functor|monoidal structure]] of $V$ is inherited by these submonads too, allowing the formation of joints and marginals.

See also [[monads of probability, measures, and valuations\#detailed\_list|monads of probability, measures, and valuations - detailed list]].

\hypertarget{normalized_valuations}{}\subsubsection*{{Normalized valuations}}\label{normalized_valuations}

If one restricts to \emph{normalized} valuations, i.e.{\tt \symbol{126}}those $\nu\in V X$ with $\nu(X)=1$, one obtains a submonad of $V$ which can be thought of as the one of [[probability]] valuations.

\hypertarget{the_measure_monad_on_top}{}\subsubsection*{{The measure monad on Top}}\label{the_measure_monad_on_top}

One can restrict $V$ only to those valuations which are [[valuation (measure theory)\#extending\_valuations\_to\_measures|extendable to measures]]. The resulting subspace $M X\subseteq V X$ (for every topological space $X$) is the space of [[tau-additive measures]] on $X$, with the subspace topology inherited by $V X$. For probability measures, this topology is sometimes known as the \textbf{A-topology}, after Alexandrov (not to be confused with the [[Alexandrov topology]], which is a different concept), for example in \hyperlink{bogachev2}{Bogachev, section 8.10.iv}. The [[specialization preorder]] is again the [[stochastic order]]. Since extendable valuations are stable with respect to pushforwards and integrations, $M$ forms a submonad of $V$, the \textbf{measure monad on [[Top]]}.

More details can be found in \hyperlink{support}{Fritz-Perrone-Rezagholi `19}, worked out explicitly for the normalized case (see below).

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

\hypertarget{the_probability_monad_on_top}{}\subsubsection*{{The probability monad on Top}}\label{the_probability_monad_on_top}

If one restricts the measure monad above to the $\tau$-smooth \emph{probability} measures (i.e. normalized), one obtains again a submonad, which seems to be the most general [[probability monad]] on [[Top]].

If a topological space is [[Tychonoff space|Tychonoff]] (for example a [[metric space]] or a [[compactum|compact Hausdorff space]]), the A-topology for probability measures coincides with the usual [[weak topology]] of measures with respect to continuous functions. In particular, on the [[subcategory]] of [[compactum|compact Hausdorff spaces]], this monad restricts to the [[Radon monad]].

\hypertarget{the_monad_of_topological_cones}{}\subsubsection*{{The monad of topological cones}}\label{the_monad_of_topological_cones}

If one restrict to [[valuation (measure theory)\#simple\_valuations|simple valuations]], i.e. those that are linear combinations of deltas, one obtains again a submonad of $V$, which can be thought of as the free [[topological cone]] monad (or free internal $[0,\infty]$-[[module object]] monad).

\hypertarget{the_monad_of_topological_convex_spaces}{}\subsubsection*{{The monad of topological convex spaces}}\label{the_monad_of_topological_convex_spaces}

If one further restricts to \emph{normalized} simple valuations, one obtains as submonad the free [[topological convex space]] monad.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

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


\item [[monads in computer science]]


\item [[valuation (measure theory)]]


\item [[correspondence between measure and valuation theory]]


\item [[Giry monad]], [[Radon monad]], [[probabilistic powerdomain]], [[valuation monad on locales]], [[distribution monad]]


\item [[convex space]], [[conical space]]



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

\begin{itemize}%
\item Olaf Kirch, \emph{Bereiche und Bewertungen} (in German), Master Thesis, Technische Hochschule Darmstadt, 1993 (\href{http://fb04286.mathematik.tu-darmstadt.de/fbereiche/logik/research/Domains/papers/kirch/diplom.ps.gz}{ps.gz})


\item Reinhold Heckmann, \emph{Spaces of valuations}, Papers on General Topology and Ap-plications, 1996 (\href{https://doi.org/10.1111/j.1749-6632.1996.tb49168.x}{doi:10.1111/j.1749-6632.1996.tb49168.x},\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5845&rep=rep1&type=pdf}{pdf})


\item Achim Jung, \emph{Stably compact spaces and the probabilistic powerspace construction}, ENTCS 87, 2004 (\href{https://doi.org/10.1016/j.entcs.2004.10.001}{doi:10.1016/j.entcs.2004.10.001}).


\item Mauricio Alvarez-Manilla, Achin Jung, [[Klaus Keimel]], \emph{The probabilistic powerdomain for stably compact spaces}, Theoretical Computer Science 328, 2004 (\href{https://doi.org/10.1016/j.tcs.2004.06.021}{doi:10.1016/j.tcs.2004.06.021})


\item [[Jean Goubault-Larrecq]] and Xiaodong Jia, \emph{Algebras of the extended probabilistic powerdomain monad}, ENTCS 345, 2019 (\href{https://doi.org/10.1016/j.entcs.2019.07.015}{doi:10.1016/j.entcs.2019.07.015})


\item [[Tobias Fritz]], Paolo Perrone and Sharwin Rezagholi, \emph{Probability, valuations, hyperspace: Three monads on Top and the support as a morphism}, 2019 (\href{https://arxiv.org/abs/1910.03752}{arXiv:1910.03752})


\item V. Bogachev, \emph{Measure Theory}, vol. 2 (2007).



\end{itemize}
[[!redirects extended probabilistic powerdomain]] [[!redirects valuation monad on top]] [[!redirects valuation monad on topological spaces]]



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