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Stream: theory: categorical probability

Topic: exponentiable measurable spaces

Matteo Capucci (he/him) (May 22 2025 at 11:11):

Is it known if there are some measurable spaces $E$ for which $X^E$ always exists? Or at least, is it known whether some $X^E$ exists? e.g. $E$ discrete works afaiu, anything else?

Morgan Rogers (he/him) (May 22 2025 at 12:06):

What are the measurable sets of $X^E$ when $E$ is uncountably infinite and discrete?

Paolo Perrone (May 22 2025 at 13:15):

It is well known that already for $2^\mathbb{R}$, the evaluation map is not measurable. [Aumann '61]

Paolo Perrone (May 22 2025 at 13:18):

It is also known that measurable spaces are monoidal closed, for a different monoidal structure than the cartesian one (giving separately and not jointly measurable functions).
However the Giry monad is not strong for that monoidal structure (the strength on uncountable spaces is not measurable). [Sato '16]

Paolo Perrone (May 22 2025 at 13:20):

Morgan Rogers (he/him) said:

What are the measurable sets of $X^E$ when $E$ is uncountably infinite and discrete?

If by $X^E$ we mean the product $\sigma$-algebra, then it is generated by sets in the form $\prod_{e\in E} A_e$ where $A_e$ is a measurable subset of $X$, and moreover $A_e=X$ for all but a countable number of $e$.

Paolo Perrone (May 22 2025 at 13:24):

This failure of exponentiability, similar to what happens with differentiable manifolds, is what leads to quasi-Borel spaces.

Matteo Capucci (he/him) (May 27 2025 at 07:38):

alas