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There are at least three different answers that can be given to this question, and in all three interpretations the answer essentially states that all maps are “open”, for the appropriate analogue of …

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 spe …

According to Proposition 416Q(b) in Fremlin's Measure Theory, finitely additive functionals A→[0,∞) are in a canonical bijective correspondence with finite Radon measures on the Stone space Spec(A) of …

The property Σ=Σ_1 amounts to (X,Ε,μ) being locally determined. A measure space (X,Σ,μ) is locally determined if μ is semifinite and A∈Σ if and only if A∩F∈Σ for all F∈Σ such that μ(F) is finite. See …

Any compact Hausdorff space $X$ is a Baire space: if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets, also known as a set of first category), then $X$ is empty. I am i …

As shown in the paper Gelfand-type duality for commutative von Neumann algebras, the following categories are equivalent. The category CSLEMS of compact strictly localizable enhanced measurable spac …

There is a complete classification of probability spaces up to a measure-preserving isomorphism. Specifically, consider a category whose objects are triples (X,Σ,μ), where X is a set, Σ is a σ-algebr …

Any smooth manifold has a canonical σ-ideal of negligible subsets, and μ must vanish on these. Apart from that, the Lie derivative of μ with respect to any smooth vector field must exist. This is ho …

Yes, it's called a pushforward! For more details, see this answer: Conditional Expectation for $\sigma$-finite measures

One can define a reasonable notion of conditional expectation for arbitrary localizable measurable spaces, not necessarily σ-finite. This is explained in great detail in the answer to Is there an intr …

This is an answer to the new question formulated in the comments. Point-set notions of topological spaces are a poor fit for arbitrary toposes because constructing points typically requires the axiom …

Take the category of measurable locales, equip it with its natural Grothendieck topology, and take the topos of sheaves of sets on the resulting site. (Apply standard disclaimers about universes, coac …

First of all, one should mention that not every triple (X,B,μ) (i.e., what is often called a measure space) satisfies the property that its C*-algebra of bounded functions is a von Neumann algebra (= …

The category of commutative von Neumann algebras is contravariantly equivalent to the category of measurable spaces. Assuming the axiom of choice, isomorphism classes of objects in the above two cate …

To define pullbacks of measures we need some additional data, because otherwise one would be able to obtain a canonical measure on an arbitrary measurable space M by pulling back the canonical measure …