# Dense subcategory of measurable spaces

Recall the notion of a dense subcategory $$\mathcal{D}$$ of a category $$\mathcal{C}$$. It means that the restricted Yoneda functor $$\mathcal{C} \to \mathrm{Hom}(\mathcal{D}^{op},\mathbf{Set})$$, $$A \mapsto \mathrm{Hom}(-,A)|_{\mathcal{D}}$$ is fully faithful. Roughly, it means that $$\mathcal{D}$$ "detects morphisms" in $$\mathcal{C}$$.

One can show that $$\mathbf{Meas}$$, the category of measurable spaces$$^1$$, has no small dense subcategory. Trivially, $$\mathbf{Meas}$$ is a dense subcategory of $$\mathbf{Meas}$$, but that is not very interesting.

Question. What is an example of a "quite small" proper dense full subcategory of $$\mathbf{Meas}$$?

By "quite small" I mean that we are not just removing a bunch of measurable spaces, but rather that the objects of the dense subcategory are parametrized by a very simple structure. Imagine, very informally, there was a measure on $$\mathbf{Meas}$$, then I want the dense subcategory to be of measure $$0$$.

We can assume that the one-point measurable space belongs to the subcategory. If $$\mathcal{K}$$ denotes the rest, we have the following characterization of density: If $$X,Y$$ are measurable spaces, then a map $$f : X \to Y$$ is measurable iff for every measurable map $$a : A \to X$$ for $$A \in \mathcal{K}$$ the composition $$f \circ a : A \to Y$$ is measurable. (This is what I meant above with "detecting morphisms"). The question asks for such a class of measurable spaces.

At first you might think that this is completely impossible. I had the same suspicion for $$\mathbf{Top}$$, but it turns out that for $$\mathbf{Top}$$ it is possible: take the one-point-space and the topological spaces of the form $$P \cup \{\infty\}$$ for directed sets $$P$$, where the sets $$P_{\geq p} \cup \{\infty\}$$ form a local base at $$\infty$$. This subcategory is dense: This is just a fancy way of saying that a map is continuous iff it preserves net convergence. Maybe there is some similar theory of "net convergence" for measurable spaces? I found the related discussion What properties are preserved under a measurable mapping?, but I am not sure if Eric Wofsey's answer settles my question, because convergent filters cannot be seen as maps.

$$^1$$ Since Dmitri Pavlov's notion of a measurable space has become quite prominent on mathoverflow, let me mention that I use the "classical" definition here. It's just a set with a $$\sigma$$-algebra. However, if there was a very good answer for Pavlov's measurable spaces, I would be happy to hear about that too.

• It is not so clear to me that this category has no dense small subcategories. Do you have a reference or argument for this ? Jun 2, 2021 at 22:21
• @Simon I haven't thought much about it, but I assume that the proof for topological spaces can be adjusted to measurable spaces, see math.stackexchange.com/a/4097368/1650. Instead of taking subsets bounded by some cardinality, we take subsets with this property whose complement also has this property. But I will check it. Jun 3, 2021 at 9:48
• Meas is a complete subcategory of the category of uniform spaces. So one could ask the same question for that larger category as well. Jun 6, 2021 at 16:54

A rather satisfying answer to this question can be given if one is willing to equip measurable spaces with a σ-ideal of negligible sets (i.e., sets of measure 0, except that we need not choose any specific measures). This is an extremely natural choice to make, since the resulting category is contravariantly equivalent to the category of commutative von Neumann algebras.

Assuming we are working in the category CSLEMS of compact strictly localizable enhanced measurable spaces described there, one can give a complete classification (essentially due to von Neumann and Maharam) of objects in CSLEMS up to an isomorphism. Specifically, any objects of CSLEMS is canonically isomorphic to the disjoint union of its atomic and nonatomic parts, and the nonatomic part is canonically isomorphic to the disjoint union of nonempty measurable spaces $$F_κ$$, where $$κ$$ is an infinite cardinal and $$F_κ$$ is noncanonically isomorphic to $$I⨯2^κ$$, where $$I$$ is an infinite set and $$2^κ$$ is interpreted as the product of $$κ$$ copies of measurable spaces $$2=\{0,1\}$$. (And for $$κ=0$$ we recover the atomic part mentioned above, so it is also covered by this construction if we allow $$I$$ to be finite in this case.)

Furthermore, the classification works in the relative case, i.e., for morphisms in CSLEMS. Indeed, mapping to a disjoint union of measurable spaces amounts to partitioning the domain and mapping each part separately. Thus, it suffices to describe maps of the form $$F_κ→F_λ$$ for some infinite cardinals $$κ$$ and $$λ$$ (we can also allow $$κ=0$$ or $$λ=0$$). Such morphisms exist if and only if $$κ≥λ$$. Furthermore, after performing a further (canonical) partition of the domain and codomain, we can make the resulting parts noncanonically isomorphic to the projections $$I⨯J⨯2^κ→I⨯2^λ$$, given by the product of the projection $$I⨯J→I$$ and the projection $$2^κ→2^λ$$.

This relative Maharam theorem allows us to easily identify dense subcategories of CSLEMS: these are precisely those subcategories that have objects with nonempty components $$F_κ$$ for arbitrary large cardinals $$κ$$. In particular, no such subcategory can be small. For example, we could take the spaces $$2^κ$$ for all infinite cardinals $$κ$$.

• Thank you, Dmitri! Do you think that there is an answer for the category of all measurable spaces? My question was directed to this category. Can you perhaps also say how restrictive the conditions "compact" and "strictly localizable" are? Are important examples of measurable spaces missing? Jun 4, 2021 at 20:01
• @MartinBrandenburg: Concerning compactness and strict localizability: all Radon measures satisfy these properties (see Example 4.55 in my paper), as well as many others. Nonlocalizable measurable spaces are extremely pathological: all nontrivial theorems from a typical measure theory textbook fail for them. In particular, the Radon–Nikodym theorem fails, the Riesz representation theorem fails, the duality theorem for L^p spaces fails, etc. So you are not doing measure theory anymore! Jun 5, 2021 at 17:28
• @MartinBrandenburg: And the category of localizable enhanced measurable spaces (with appropriately defined morphisms) is equivalent to the category of compact strictly localizable enhanced measurable spaces (Remark 5.18 in my paper), although the latter is more convenient by having a simple point-set description of its morphisms. Jun 5, 2021 at 17:30
• Thanks! Which paper of yours do you mean here? Jun 5, 2021 at 17:35
• @MartinBrandenburg: I meant the paper cited in my answer: arxiv.org/abs/2005.05284. Jun 5, 2021 at 17:37