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February 4, 2025

A Characterization of Standard Borel Spaces

Posted by John Baez

People in measure theory find it best to work with, not arbitrary measurable spaces, but certain nice ones called standard Borel spaces. I’ve used them myself.

The usual definition of these looks kind of clunky: a standard Borel space is a set XX equipped with a σ\sigma-algebra Σ\Sigma for which there exists a complete metric on XX such that Σ\Sigma is the σ\sigma-algebra of Borel sets. But the results are good. For example, every standard Borel space is isomorphic to one of these:

  • a finite or countably infinite set with its σ\sigma-algebra of all subsets,
  • the real line with its sigma-algebra of Borel subsets.

So standard Borel spaces are a good candidate for Tom Leinster’s program of revealing the mathematical inevitability of certain traditionally popular concepts. I forget exactly how he put it, but it’s a great program and I remember some examples: he found nice category-theoretic characterizations of Lebesgue integration, entropy, and the nerve of a category.

Now someone has done this for standard Borel spaces!

This paper did it:

Here is his result:

Theorem. The category SBor\mathsf{SBor} of standard Borel spaces and Borel maps is the (bi)initial object in the 2-category of countably complete countably extensive Boolean categories.

Here a category is countably complete if it has countable limits. It’s countably extensive if it has countable coproducts and a map

X→∑ iY i X \to \sum_i Y_i

into a countable coproduct is the same thing as a decomposition X≅∑ iX iX \cong \sum_i X_i together with maps X i→Y iX_i \to Y_i for each ii. It’s Boolean if the poset of subobjects of any object is a Boolean algebra.

So, believe it or not, these axioms are sufficient to develop everything we do with standard Borel spaces!

Posted at February 4, 2025 6:13 PM UTC

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Re: A Characterization of Standard Borel Spaces

This is so unexpected! I would have guessed the initial such category to be the category of countable sets. (Analogously to how the initial Boolean lextensive category is the category of finite sets).

In fact, I still don’t see how the category of countable sets fails to be initial: it is generated under countable coproducts by the terminal object, so there seems to be exactly one choice (up to iso) for defining a functor preserving countable limits (hence the terminal object) and countable coproducts into any “countably lextensive” category. What am I missing?

Posted by: Damiano Mazza on February 4, 2025 8:21 PM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

The category of countable sets doesn’t have countable limits: ∏ ℕℕ\prod_{\mathbb{N}} \mathbb{N} isn’t countable.

Perhaps the category of countable sets is the initial finitely complete countably extensive Boolean category?

Posted by: Mike Shulman on February 4, 2025 10:05 PM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

Of course!!! It had to be something silly like that :-) Thanks.

Posted by: Damiano Mazza on February 5, 2025 6:34 AM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

Thanks for saving the day, Mike!

By the way, this paper relies on something called Loomis–Sikorski duality, which says that the category of standard Borel spaces is the opposite of the category of countably presented countably complete Boolean algebras. I presume this result is used to put the concept of standard Borel space into a more algebraic framework, making it easier to prove the desired universal property.

Posted by: John Baez on February 5, 2025 12:28 AM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

This is really cool! I like to talk (section 2.8) about how even if we start out just working with sets, certain constructions “automatically” force us to think about some kind of space instead. The most important constructions that do that are (1) infinite limits, (2) function-spaces, and (3) universes of sets or propositions. So this theorem can be interpreted as saying that if the only such space-making construction we have is countably infinite limits (since coproducts and finite limits don’t take us out of discrete sets), then measurable spaces are the “universal” notion of “space” we get, and moreover up to isomorphism there’s only one non-discrete space we get.

Posted by: Mike Shulman on February 5, 2025 1:41 AM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

if the only such space-making construction we have is countably infinite limits (since coproducts and finite limits don’t take us out of discrete sets), then

I should have added “and if we insist on retaining classical logic”.

Now I wonder what the analogous universal objects are if we replace “Boolean category” by “coherent category” or “Heyting category”.

I also wonder what the internal logic of the category of all measurable spaces is like. Presumably it is still Boolean. Is it cartesian closed?

Posted by: Mike Shulman on February 5, 2025 2:00 AM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

Meas is not cartesian closed. Robert Furber’s answer at the link gives the arguably cleanest way to see that: if YY is any discrete measurable space of cardinality |Y|≥|ℝ||Y| \ge |\mathbb{R}|, then the functor −×Y- \times Y does not preserve coproducts.

In fact, Meas is not a good category for the purposes of probability theory, similar to how Top is not a good category for the purposes of homotopy theory. SBor (=BorelMeas) works well for most practical purposes, but only contains the “small” spaces.

Finding a convenient category of measurable or measurable-like spaces that works for probability is hard. Quasi-measurable spaces don’t work, because in probability it should be the case that every measure on a product space X×YX \times Y which has deterministic marginal on XX is a product measure, and quasi-measurable spaces fail this (see Prop 3.3 here). Forré’s quasi-measurable spaces seem to give a convenient category that works, but the definitions are a bit unwieldy.

Posted by: Tobias Fritz on February 5, 2025 7:46 AM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

Typo: I meant to say that quasi-Borel spaces don’t work. Forré’s quasi-measurable spaces are different and apparently do work. That being said, I still have hope that a nicer convenient category exists.

Posted by: Tobias Fritz on February 5, 2025 7:50 AM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

Mike wrote:

So this theorem can be interpreted as saying that if the only such space-making construction we have is countably infinite limits (since coproducts and finite limits don’t take us out of discrete sets), then measurable spaces are the “universal” notion of “space” we get, and moreover up to isomorphism there’s only one non-discrete space we get.

I’d like to know a similar characterization of Meas, the category of all measurable spaces. I might sneak up on it by trying to generalize Loomis–Sikorski duality, which says that the category of standard Borel spaces is the opposite of the category of countably presented countably complete Boolean algebras. Any measurable space has a countably complete Boolean algebra of measurable subsets. ‘Countably generated’ is a size condition, as is the definition of standard Borel space. So I might guess that the category of measurable spaces is the opposite of the category of countably complete Boolean algebras.

Even if this guess is wrong, the opposite of the category of countably complete Boolean algebras should be some interesting category of spaces!

Posted by: John Baez on February 5, 2025 8:19 AM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

So I might guess that the category of measurable spaces is the opposite of the category of countably complete Boolean algebras.

That’s a good guess, but it’s not quite right: there are countable complete Boolean algebras which cannot be represented as a σ-algebra.

Even if this guess is wrong, the opposite of the category of countably complete Boolean algebras should be some interesting category of spaces!

Well, one can apply Stone duality: those Stone spaces that correspond to countably complete Boolean algebras are known as Rickart spaces. At the level of morphisms, there are two choices: do you allow all Boolean algebra morphisms, or only those which also preserve the countable infima? The latter is perhaps more natural, and this imposes a further restriction on the maps between Rickart spaces. So there is a dual equivalence, but it’s not particularly nice and it’s basically designed to be the obvious restriction of Stone duality. And I don’t think that one can do better than this due to the above failure of representability.

On the other hand, one can prove that the category of countably complete Boolean algebras is equivalent to the category of ‘countably complete’ commutative C*-algebras, where ‘countably complete’ means that countable suprema of bounded sequences of self-adjoint elements exist.

So there’s a plethora of categories connected by equivalences, adjunctions at play here, some of them covariant and some contravariant. Antonio Lorenzin and myself will soon have a preprint out on all of this (on which Antonio has done a good deal more than me). If I could paste images here, I’d include our big diagram that illustrates all of these categories and adjunctions. We hope that it can be useful for finding a new convenient category for measure-theoretic probability, perhaps even in the quantum case.

Posted by: Tobias Fritz on February 5, 2025 8:38 AM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

I should add that Foundational aspects of uncountable measure theory by Jamneshan and Tao is a recent paper that is closely related. For example, they consider one version of the dual equivalence that I’ve described for countably complete Boolean algebras in Proposition 9.2(iv).

Posted by: Tobias Fritz on February 5, 2025 12:48 PM | Permalink | Reply to this

Re: A Characterization of Standard Borel Spaces

Thanks for all that information, Tobias.

The “mathematical inevitability” programme suggests that maybe we ought to be looking at something like the initial objects in the categories of κ\kappa-complete countably extensive Boolean categories, for uncountable cardinals κ\kappa.

(I’d like to take κ=Ord\kappa=Ord, but in that case it’s not clear that there is an initial object, for size reasons. And one could wonder about whether the size of extensivity should also go up; my initial guess is not, just by analogy to how σ\sigma-algebras are only countably additive no matter how large they are.)

Posted by: Mike Shulman on February 6, 2025 4:57 PM | Permalink | Reply to this

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