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.
2026-04-29 13:07:57.1777468077
Examples of universal constructions in probability theory
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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.