Examples of universal constructions in probability theory
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.
Cylinder set measures are defined categorically if not universally, and are used to define measures on infinite-dimensional spaces such as the abstract wiener space construction.
Lebesgue measure and the integral can be defined universally as shown by Tom Leinster.