nLab
Radon monad

Contents

Contents

Idea

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

There is also an ordered version of the Radon monad on the category of compact ordered spaces (see below).

Spaces of Radon measures

Let XX be a compact Hausdorff space. Denote by RXR X the set of those Radon measures on XX which are subnormalized, i.e. those measures μ\mu for which μ(X)1\mu(X)\le 1. Denote by PXP X its subset of Radon probability measures.

Equip both sets with the topology of weak convergence with respect to continuous functions. It is well known that the resulting spaces RXR X and PXP X are themselves compact Hausdorff spaces.

In terms of τ-additive measures

Since on compact Hausdorff spaces Radon and τ-additive measures coincide, we can equivalently define RXR X and PXP X as the sets of subnormalized and normalized τ-additive measures.

For probability measures, the weak topology and the A-topology coincide, so that PP an be extended to the probability monad on Top.

The same cannot be said for RR, since the weak topology and the A-topology do not coincide for subnormalized measures: given μRX\mu\in R X, the measure rμr \mu for 0r10 \le r \le 1 is in the closure of μ\mu for the AA-topology, while the weak topology is Hausdorff.

In terms of valuations

Note that, since every continuous valuation on a compact Hausdorff space can be 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 pointwise topology of valuations coincide, so that PP an be extended to a submonad of the extended probabilistic powerdomain on Top. Again, the same cannot be said for RR.

Functoriality, unit and multiplication

Let f:XYf:X\to Y be a continuous map between compact Hausdorff spaces. The pushforward of measures (or equivalently of valuations) gives a well-defined, continuous map RXRYR X\to R Y which restricts to the map PXPYP X \to P Y. This makes RR and PP 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 XX, we can assign to each xXx\in X its Dirac measure (equivalently, Dirac valuation δ x\delta_x. The assignment δ:XPX\delta:X\to P X, or XRXX\to R X is continuous, and natural in XX.

Just as well, given a measure μPPX\mu\in P P X, we can define the measure EμPXE\mu\in P X as the one assigning to a measurable AXA\subseteq X the number

Eμ(A) PXp(A)dμ(p). E\mu(A) \coloneqq \int_{P X} p(A) \,d\mu(p).

(In terms of valuations, the same is given for open sets instead of measurable sets.) Again, the assignment E:PPXPXE: P P X \to P X is continuous and natural in XX. The multiplication for RR is defined analogously.

The unit and multiplication thus defined satisfy the usual axioms of a monad. The monads RR and PP are both known in the literature as the Radon monad.

The monad PP is the restriction to compact Hausdorff spaces of the probability monad on Top, which is itself a submonad of the extended probabilistic powerdomain. (See also monads of probability, measures, and valuations.)

Algebras

The algebras of the Radon monad PP are compact convex subsets of locally convex topological vector spaces.

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

p Ccdp(c). p \mapsto \int_{C} c \,dp(c) .

Since the space is compact, the integral above is well-defined, and it returns an element of CC which we can view as the “center of mass” of pp.

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

The 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 affine maps, i.e. those maps f:ABf:A\to B which satisfy

f(λa+μb)=λf(a)+μf(b) f\big( \lambda\, a + \mu\, b \big) \;=\; \lambda\,f(a) + \mu\,f(b)

for all a,bAa,b\in A and 0λ,μ10\le\lambda,\mu\le 1 (with λ+μ=1\lambda+\mu=1 for the normalized case).

For more information, see Swirszcz ‘74 and the later Keimel ‘08.

The ordered case

The Radon monad can be lifted to the category of compact ordered spaces (and continuous monotone maps). This is done by means of the stochastic order.

Here we sketch the construction. For more details, see Keimel ‘08.

Construction of the spaces

Given a compact ordered space XX, we can form the spaces RXR X and PXP 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 RXR X and PXP X are again compact ordered spaces.

Given a monotone map f:XYf:X\to Y, the maps RfR f and PfP f are monotone for the stochastic order, and so are the monad structure maps. This way, PP and RR lift to a monad on CompOrd.

For the canonical locally posetal 2-category structure of CompOrd given by the pointwise order, PP and RR are even 2-monads, since whenever fg:XYf\le g:X\to Y, we have RfRgR f\le R g PfPgP f\le P g.

Both the resulting monads are known in the literature as ordered Radon monad.

Algebras

The algebras of the ordered Radon monad PP are known to be compact convex subsets of locally convex ordered topological vector spaces?. This can be seen as the ordered equivalent of the characterization above.

The algebras of RR, similarly, can be seen as the ordered equivalent of Kegelspitze?.

See Keimel ‘08 for more.

Just as for the unordered case, the algebra morphisms are the continuous affine maps, which here are also required to be monotone.

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:ABf:A\to B with the property that

f(adp(a))f(a)dp(a) f \left( \int a \, dp(a) \right) \;\le\; \int f(a) \, dp(a)

for all pPAp\in P A (resp. RAR A).

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

f(λa+μb)λf(a)+μf(b). f\big( \lambda\, a + \mu\, b \big) \;\le\; \lambda\,f(a) + \mu\,f(b) .

(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 F-P ‘18.)

See also

References

  • T. Swirszcz, Monadic functors and convexity, Bulletin de l’Academie Polonais des Sciences 22, 1974 (pdf)

  • Klaus Keimel?, The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras, Topology and its Applications, 2008 (doi:10.1016/j.topol.2008.07.002)

  • Tobias Fritz and Paolo Perrone, Stochastic order on metric spaces and the ordered Kantorovich monad, submitted, 2018. (arXiv:1808.09898)

Last revised on October 25, 2019 at 21:09:35. See the history of this page for a list of all contributions to it.