nLab Radon monad

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 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 $X$ be a compact Hausdorff space. Denote by $R X$ the set of those Radon measures on $X$ which are 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 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.

In terms of τ-additive measures

Since 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 A-topology coincide, so that $P$ an be extended to the probability monad on Top.

The same cannot be said for $R$, since the weak topology and the A-topology do not coincide for 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.

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 $P$ an be extended to a submonad of the extended probabilistic powerdomain on Top. Again, the same cannot be said for $R$.

Functoriality, unit and multiplication

Let $f:X\to Y$ be a continuous map between compact Hausdorff spaces. The pushforward of measures (or equivalently of valuations) gives a well-defined, continuous map $R X\to R Y$ 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, Dirac valuation $\delta_x$. The assignment $\delta:X\to P X$, or $X\to R X$ is continuous, and 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

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

(In terms of valuations, the same is given for open sets instead of measurable sets.) Again, the assignment $E: P P X \to P X$ is continuous and 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 Radon monad.

The monad $P$ 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 $P$ 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:

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

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 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 $f:A\to B$ which satisfy

$$f\big( \lambda\, a + \mu\, b \big) \;=\; \lambda\,f(a) + \mu\,f(b)$$

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 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 $X$, 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.

For the canonical locally posetal 2-category structure of 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 ordered Radon monad.

Algebras

The algebras of the ordered Radon monad $P$ 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 $R$, 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 $f:A\to B$ with the property that

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

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

$$f\big( \lambda\, a + \mu\, b \big) \;\le\; \lambda\,f(a) + \mu\,f(b) .$$

(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

• monads of probability, measures, and valuations

• Giry monad, extended probabilistic powerdomain, Kantorovich monad

• locally convex topological vector space, ordered vector space?, convex space

• compact Hausdorff space, compact ordered space, stably compact space

• Borel measure, Radon measure, τ-additive measure, continuous valuation

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, Advances in Mathematics 366, 2020. (arXiv:1808.09898)