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Consider the category with the standard Lebesgue measure space $\Omega$ as its only object and measure type preserving nonincreasing (equivalence classes of) maps as morphisms.

Question: is there an account of measure theory on the topos of sheaves on this site (with the obvious notion of countable coverings), or any similar kind of topos of "generalized measure type spaces"?

My motivation for this question is purely probabilistic. There are many situations when we encounter objects of this topos that are not standard measure spaces, and probabilistic reasoning (e.g. things as basic as disintegration) becomes problematic or nontrivial there. Some examples include:

  1. Categorical products of measure spaces.

  2. Singular factors, like $\mathbb{R}/\mathbb{Q}$, or, for instance, the space of dense countable sets (the latter was studied by Tsirelson).

  3. To every (say) Hilbert space $H$ we may associate the "space" $\bar{H}$, such that "random vectors" in that space are continuous operators $H \to L^0(\Omega)$. For example, the "standard Gaussian random vector in $H$" is a bona fide random vector in $\bar{H}$. Even to state the basic fact that $\bar{H}$ is a vector space we already need some rudimentary categorical language. For an example of a useful and analytically nontrivial theorem about such vectors, see the answer to this question of mine.

  4. Related to the previous one are Skorokhod's "random operators", Tsirelson's "S-maps", and many more.

P.S. There may be variations on the category, like, say, replacing maps by Markovian kernels, or even allowing both at the same time (that would require two objects, one "randomized" and one that is not), or allowing other measure spaces like spectra of von Neumann algebras over non-separable Hilbert space, commutative or maybe even not. I'm not sure what is "the" best one...

P.P.S. I know that there are approaches to measure theory over generalized "Borel spaces", "compactly generated spaces" or locales, but I'd like to emphasize a point of view that descriptive-set-theoretic aspects of measure theory (i.e. working with sets as opposed to measures and equivalence classes of sets) are often irrelevant for probability, which is why I want the spaces tested by a Lebesgue-like structure, not Borel or topological.

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I am not entirely sure what you are looking for, but this may be relevant A Sheaf Theoretic Approach to Measure Theory.

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If you are looking for measure theory in a topos, this paper by Vickers on valuations may be more relevant.

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