Obviously certains construction as for instance the push-forward (direct image) ${\cdot}_{\star}$ can be interpreted in the categories of measurable spaces, measured spaces or probability spaces as universal in the sense that they represent a functor.
Nevertheless universality of such constructions is not really interesting, as the aforementioned categories themselves are not very interesting, as measure theory is not about the category of measurable spaces with measurable maps nor is probability theory about probability spaces and measurable maps as, for instance, isomorphic objects in the latter category are not the same from the point of view of probability theory. (Think for example of the uniform distribution on $[0,1[$ and a countably infinite sequence of independent coin tosses.)
Rather, I would be interested in for instance
- the Lebesgue measure (or more generally any Haar measure of a locally compact abelian topological group)
- the Brownian motion representing a functor (the Wiener space feels somehow universal to me)
- any given "distribution"
representing a functor. (I secretely hope that these functors would also allow to explain that even if "we do not care about the sample space", some sample spaces look more universal an natural than others ...)
Do I miss the obvious ? If not, are there references about this questions of question somewhere ?