In which category is a measure on a measurable space a morphism?

Question

I'd like to be able to say that a measure $\mu$ on a measurable space $X$ "is" a morphism $R \to X$, where $R$ is some incarnation of the real numbers in an appropriate category.

In other words, let $Mbl$ be the category of measurable spaces (an object is a set $X$ equipped with a $\sigma$-algebra; a morphism $f : X \to Y$ is a measurable function, i.e. the preimage of a measurable set is measurable). There is a functor $Meas : Mbl \to Set$, carrying $X$ to the set of measures on $X$.

Question: Is there some natural enlargement $\widetilde{Mbl}$ of the category $Mbl$ such that the functor $Meas : Mbl \to Set$ extends to a corepresentable functor $\widetilde{Mbl} \to Set$?

In other words, is there another category $\widetilde{Mbl}$, a fully faithful functor $\iota: Mbl \to \widetilde{Mbl}$, an object $R \in \widetilde{Mbl}$, and a natural isomorphism $Meas \cong Hom_{\widetilde{Mbl}}(R, \iota - )$?

(Of course, the answer is "trivially yes", but the point is that $\widetilde{Mbl}$ should be "natural" in some imprecise sense, ruling out the trivial answer.)

I'd also be happy for an answer where some slight modification of the category $Mbl$ is used. For example, perhaps we need to equip our measure spaces with an ideal of negligible sets or something...

Answer 1

How about this? (But I will be enlarging $Mbl$ by a functor that is not full.)

Let's make a category whose objects can be called abstract $\sigma$-algebras. I'll call the category $\Sigma$.

A $\Sigma$-object consists of

(1) a set $F$,

(2) a symmetric binary relation on $F$ called "disjointness", and

(3) a rule "countable disjoint union" that assigns to each countable family $x:S\to F$ of pairwise disjoint elements an element $D(x)\in F$. (Here $S$ is a finite or countably infinite set, and "pairwise disjoint" means that $x(i)$ and $x(j)$ are disjoint if $i\neq j$.)

This is subject to some axioms. Call $x:S\to F$ and $x':S'\to F$ disjoint if $x(i)$ is always disjoint from $x'(i')$. The axioms are as follows:

(A1) If $E\in F$ is disjoint from $x(i)$ for every $i\in S$ then $E$ is disjoint from $D(x)$. Note that A1 implies that if $x:S\to F$ and $x':S'\to F$ are disjoint then $D(x)$ and $D(x')$ are disjoint.

(A2) For $x:\lbrace i_0\rbrace \to F$, $D(x)=x(i_0)$.

(A3) The disjoint union of disjoint unions is the disjoint union.

A $\Sigma$-morphism $T:F\to F'$ is a map of sets from $F$ to $F'$ that preserves disjointness and commutes with disjoint union in the sense that:

(M1) Whenever $E_1$ and $E_2$ are disjoint elements of $F$ then $T(E_1)$ and $T(E_2)$ are disjoint.

(M2) When $x:S\to F$ is a countable pairwise disjoint family (i.e. when $D(x)$ is defined) then $D(T\circ S))$ (which is defined, by (M1)) is equal to $T(D(S))$.

There is a functor $\iota:Mbl \to \Sigma^{op}$, taking each measurable space $X$ to the set $M_X$ of all measurable subsets of $X$ (with the usual notions of disjointness and countable disjoint union) and taking a measurable map $X\to Y$ to the map $M_Y\to M_X$ given by $E\mapsto f^{-1}(E)$.

The functor $\iota$ is not faithful, but it becomes faithful if we restrict attention to those measurable spaces in which points are determined by which measurable sets they belong to, which is not a big deal.

Here is the desired representing object. Let $P$ be the set $[0,+\infty]$, declare any two elements to be disjoint, and for a countable family $S\to P$ let $D(x)$ be the sum.

So a $\Sigma$-morphism $F\to P$ is simply a rule assigning an element of $[0,+\infty]$ to each element of $F$ and satisfying additivity for finite and countably infinite (abstract) disjoint unions. In particular a $\Sigma^{op}$-morphism $P\to\iota(X)$ is precisely a measure on the measurable space $X$.

Note that the structure on a set (abstract disjointness relation and abstract disjoint union) that makes it an abstract $\sigma$-algebra brings with it some things. There is an element $0=D(\emptyset)$ that is disjoint from all elements. There is a weak partial ordering: $E'\le E$ iff there exist $E''$ disjoint from $E'$ such that $E$ is the disjoint union of $E'$ and $E''$.

The functor $\iota$ is not full, but some of the new morphisms between measure spaces that appear are things that one might want to use sometimes. If $X$ is a measurable space and $A\subset X$ is a measurable subset, which we then consider as a measurable space in its own right, then there is a "wrong-way" map $\iota(X)\to \iota(A)$ that takes any measurable subset $E\subset A$ to $E\subset X$.

The object $\iota(\emptyset)$ is both initial and terminal.

Answer 2

A very natural enlargement of the category of measurable spaces is $\mathsf{Stoch}$, the category of measurable spaces and Markov kernels, with composition given by the Chapman-Kolmogorov equation. This is equivalently the Kleisli category of the Giry monad, see Lawvere 1962 and Giry 1980. By construction, this has the property that the morphisms $1 \to X$ are exactly the probability measures on $X$.

I believe that this still works out just the same if you drop the normalization of probability and consider arbitrary measures instead (though I haven't worked out the details). An interesting intermediate choice could be that of s-finite measures and kernels.

As for why one might one way to consider categories with Markov kernels as morphisms, see this other answer.

---

As a technical detail, the functor $\mathsf{Mbl} \to \mathsf{Stoch}$ is not faithful, and because of this it perhaps doesn't qualify as an "enlargement" of $\mathsf{Mbl}$ in the sense of the OP. However, it is faithful on morphisms between (e.g.) standard Borel spaces, so restricting both categories to those objects should satisfy your desiderata.