nLab extended probabilistic powerdomain

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 Kirch ‘93, in German, or AJK ‘06, in English). It was extended to all of Top and given its current form by Heckmann (see Heckmann ‘96).

Definitions

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 generated by the sets in the form

$$\theta(U,r) \coloneqq \{ \nu \in V X \,|\, \nu(U) \gt r \},$$

for $r\ge 0$ and $U\subseteq X$ open, or equivalently by the sets in the form

$$\Theta(g,r) \coloneqq \left\{ \nu \in V X \,\Big|\, \int g \,d\nu \gt r \right\}$$

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 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

$$\nu \mapsto \nu(U)$$

for every open $U\subseteq X$, or

$$\nu \mapsto \int g \, d\nu$$

for every lower semicontinuous $g:X\to[0,\infty]$, are lower semicontinuous. Lower semicontinuity, in some sense, plays the role that 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 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.

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 pushforward along $f$.

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

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 Dirac valuation at $x$. This map $\delta$ is continuous, and natural in $X$.

The multiplication map makes use of the concept of 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

$$E \xi (U) \coloneqq \int_X \nu(U) \, d\xi(\nu).$$

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 and natural in $X$.

The maps $\delta$ and $E$ satisfy the usual axioms of a monad. The monad $(V,\delta,E)$ is usually called the 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.

Monoidal structure

(…)

Algebras

(…)

Notable submonads

There are a number of monads that can be constructed as submonads of $V$. The 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.

Normalized valuations

If one restricts to normalized valuations, i.e.~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.

The measure monad on Top

One can restrict $V$ only to those valuations which are 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 A-topology, after Alexandrov (not to be confused with the Alexandrov topology, which is a different concept), for example in 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 measure monad on Top.

More details can be found in Fritz-Perrone-Rezagholi ‘19, worked out explicitly for the normalized case (see below).

See also correspondence between measure and valuation theory.

The probability monad on Top

If one restricts the measure monad above to the $\tau$-smooth 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 (for example a metric space or a 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 compact Hausdorff spaces, this monad restricts to the Radon monad.

The monad of topological cones

If one restrict to 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).

The monad of topological convex spaces

If one further restricts to normalized simple valuations, one obtains as submonad the free topological convex space? monad.

Related concepts

• monads of probability, measures, and valuations

• monads in computer science

• valuation (measure theory)

• correspondence between measure and valuation theory

• Giry monad, Radon monad, probabilistic powerdomain?, valuation monad on locales?, distribution monad

• convex space, conical space

References

• Olaf Kirch, Bereiche und Bewertungen (in German), Master Thesis, Technische Hochschule Darmstadt, 1993 (ps.gz)

• Reinhold Heckmann, Spaces of valuations, Papers on General Topology and Ap-plications, 1996 (doi:10.1111/j.1749-6632.1996.tb49168.x,pdf)

• Achim Jung, Stably compact spaces and the probabilistic powerspace construction, ENTCS 87, 2004 (doi:10.1016/j.entcs.2004.10.001).

• Mauricio Alvarez-Manilla, Achim Jung, Klaus Keimel, The probabilistic powerdomain for stably compact spaces, Theoretical Computer Science 328, 2004 (doi:10.1016/j.tcs.2004.06.021)

• Jean Goubault-Larrecq and Xiaodong Jia, Algebras of the extended probabilistic powerdomain monad, ENTCS 345, 2019

  (doi:10.1016/j.entcs.2019.07.015)

• Tobias Fritz, Paolo Perrone and Sharwin Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism, 2019 (arXiv:1910.03752)

• V. Bogachev, Measure Theory, vol. 2 (2007).