I am looking for more examples of universal constructions in probability theory. Like the construction a of Gaussian space from a real Hilbert space, or a Poisson jump process from a measurable space with a $\sigma$-finite measure. There must be tons of examples, even though their universality (in the sense of category theory) is probably not commonly emphasized.
The category of measurable spaces is topological this means that they have initial & final structures analogously to those in topology. These can be universally expressed as noted on the wikipedia page.
Lebesgue measure and the integral can be defined universally as shown by Tom Leinster.