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$\def\Spec{\mathop{\rm Spec}} \def\R{{\bf R}} \def\Ep{{\rm E}^+} \def\L{{\rm L}} \def\EpL{\Ep\L}$ One can argue that an object of the right category of spaces in measure theory is not a set equipped w …

A conceptual answer can be given in the framework of this answer. The functor that sends a measurable space X to the set of random variables on X, i.e., equivalence classes of (unbounded) real or com …

From the categorical viewpoint there is no difference, because the category of measurable spaces is equivalent to the category of complete measurable spaces with the equivalence given by the completio …

I am not sure how far you want to go, but some basics are explained in this answer: Is there an introduction to probability theory from a structuralist/categorical perspective? In particular, you have …

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 …

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 …

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