nLab extended probabilistic powerdomain




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).


Spaces of valuations

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

θ(U,r){νVX|ν(U)>r}, \theta(U,r) \coloneqq \{ \nu \in V X \,|\, \nu(U) \gt r \},

for r0r\ge 0 and UXU\subseteq X open, or equivalently by the sets in the form

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

for r0r\ge 0 and g:X[0,]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

νν(U) \nu \mapsto \nu(U)

for every open UXU\subseteq X, or

νgdν \nu \mapsto \int g \, d\nu

for every lower semicontinuous g:X[0,]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 XX is stably compact, then VXV X is stably compact too, so that the monad VV restrict to the subcategory of stably compact spaces.


Given a continuous map f:XYf:X\to Y, define the map f:VXVY f:V X\to V Y as the one assigning to a continuous valuation νVX\nu\in V X its pushforward along ff.

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

Unit and multiplication

We can define the unit of the monad as follows. Given a space XX, define the map δ:XVX\delta:X\to V X as the one assigning to the point xXx\in X the Dirac valuation at xx. This map δ\delta is continuous, and natural in XX.

The multiplication map makes use of the concept of integration over a valuation. Given a valuation ξVVX\xi\in V V X, we can define the valuation EξVXE \xi\in V X as the one mapping an open set UXU\subseteq X to

Eξ(U) Xν(U)dξ(ν). E \xi (U) \coloneqq \int_X \nu(U) \, d\xi(\nu).

This integral is well defined, since the assignment Uν(U)U\mapsto \nu(U) is lower semicontinuous. The assignment UEξ(U)U\mapsto E \xi (U) gives a continuous valuation on XX, and the resulting map E:VVXVXE: V V X \to V X is continuous and natural in XX.

The maps δ\delta and EE satisfy the usual axioms of a monad. The monad (V,δ,E)(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




Notable submonads

There are a number of monads that can be constructed as submonads of VV. The monoidal structure of VV 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 νVX\nu\in V X with ν(X)=1\nu(X)=1, one obtains a submonad of VV which can be thought of as the one of probability valuations.

The measure monad on Top

One can restrict VV only to those valuations which are extendable to measures. The resulting subspace MXVXM X\subseteq V X (for every topological space XX) is the space of tau-additive measures on XX, with the subspace topology inherited by VXV 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, MM forms a submonad of VV, 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 τ\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 VV, which can be thought of as the free topological cone? monad (or free internal [0,][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.


Last revised on January 15, 2020 at 04:30:44. See the history of this page for a list of all contributions to it.