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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.

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@Martin: Thanks! –  UwF Feb 6 '13 at 8:43
I'm not sure that is what you are interested, but there has been much work on projective limits of probability spaces, starting with work of Bochner, as a genralization of the Daniell-Kolmogorov-construction of stochastic processes. –  Michael Greinecker Feb 6 '13 at 14:20
I was thinking of examples that could convince classical probabilists that it might be worthwile for them to study some basic category theory... but I might be dreaming ;) –  UwF Feb 6 '13 at 15:39
It is clearly not an answer, but it is related : a categorical approach to measure theory. –  Pece Dec 7 '13 at 14:40

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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.

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The notes by Leinster are cool, which book of Barr and Wells does he refer to? How technical does the construction of the initial object get, if one fills in all the details? –  UwF Feb 12 '13 at 9:03
The notes by Leinster cannot be found. Please correct the url. –  Ehsan M. Kermani Dec 3 '13 at 7:54
I think this is the right link maths.ed.ac.uk/~tl/pssl/leinster.pdf –  UwF Dec 3 '13 at 17:43

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