160

156

The title really is the question, but allow me to explain.

I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of Kolmogorov, i.e., probability measures) are appealing and potentially useful to me. It seems to me that, perhaps more than most other areas of mathematics, there are many, many nice introductory (as well as not so introductory) texts on this subject.

However, I haven't found any that are written from what it is arguably the dominant school of thought of contemporary mainstream mathematics, i.e., from a structuralist (think Bourbaki) sensibility. E.g., when I started writing notes on the texts I was reading, I soon found that I was asking questions and setting things up in a somewhat different way. Here are some basic questions I couldn't stop from asking myself:

[0) Define a Borel space to be a set $X$ equipped with a $\sigma$-algebra of subsets of $X$. This is already not universally done (explicitly) in standard texts, but from a structuralist approach one should gain some understanding of such spaces before one considers the richer structure of a probability space.]

1) What is the category of Borel spaces, i.e., what are the morphisms? Does it have products, coproducts, initial/final objects, etc? As a significant example here I found the notion of the product Borel space -- which is exactly what you think if you know about the product topology -- but seemed underemphasized in the standard treatments.

2) What is the category of probability spaces, or is this not a fruitful concept (and why?)? For instance, a subspace of a probability space is, apparently, not a probability space: is that a problem? Is the right notion of morphism of probability spaces a measure-preserving function?

3) What are the functorial properties of probability measures? E.g., what are basic results on pushing them forward, pulling them back, passing to products and quotients, etc. Here again I will mention that product of an arbitrary family of probability spaces -- which is a very useful-looking concept! -- seems not to be treated in most texts. Not that it's hard to do: see e.g.

http://math.uga.edu/~pete/saeki.pdf

I am not a category theorist, and my taste for how much categorical language to use is probably towards the middle of the spectrum: that is, I like to use a very small categorical vocabulary (morphisms, functors, products, coproducts, etc.) as often as seems relevant (which is very often!). It would be a somewhat different question to develop a truly categorical take on probability theory. There is definitely some nice mathematics here, e.g. I recall an arxiv article (unfortunately I cannot put my hands on it at this moment) which discussed independence of events in terms of tensor categories in a very persuasive way. So answers which are more explicitly categorical are also welcome, although I wish to be clear that I'm not asking for a categorification of probability theory *per se* (at least, not so far as I am aware!).

3

A beautiful, structural account of Borel spaces and measurable mappings is given in: http://www.ma.utexas.edu/mp_arc/c/02/02-156.pdf The category of Borel spaces and measurable mappings is sometimes denoted by meas (for measurable spaces, the nowadays more common name for Borel spaces), but the only real application I know of is in epistemic game theory, see for example chapter 7 here: https://scholarworks.iu.edu/dspace/bitstream/handle/2022/7065/umi-indiana-1146.pdf?sequence=1

– Michael Greinecker – 2012-01-01T00:55:06.5471@Michael and @Pete I am bit confused with the use of

`Borel space`

for`measurable space`

and not for a Borel subset of a Polish space. Is it a usual convention? – Ilya – 2012-03-31T20:06:01.107@Ilya: I believe this use of "Borel space" is a reasonably standard convention, although of course it is not the only one. – Pete L. Clark – 2012-04-06T03:36:56.570

1

I am certainly not an expert, but I was looking for a similar thing, and found Dudley's book (http://books.google.com/books?id=Wv_zxEExK3QC&lpg=PP1&dq=dudley%20probablity&pg=PA259#v=onepage&q&f=false) promising. He doesn't mention categories at all, but it seems that he has them in mind. In particular, he defined "measurable function" between any two measurable spaces (p. 116), [which is different from the definition in Rudin]. Also, while he proves the existence of

– user2734 – 2010-04-08T16:16:48.820countableproducts of probability spaces, he does remark on converting the proof to an arbitrary product (p. 259).4

This is not developed enough to be a (partial) answer rather than a comment, but see perhaps: http://golem.ph.utexas.edu/category/2007/02/category_theoretic_probability_1.html (and other google/Mathscinet results for "Giry monad")

– Yemon Choi – 2010-04-08T16:16:50.9476One thing I thought I'd mention - as a probablist manqué - is a comment at the beginning of Williams' Probability with Martingales, where he says something along lines of "it would be nice if we could think of random variables as equivalence classes of functions rather than functions, so that we don't need to keep inserting 'a.e.' everywhere; but this point of view runs into trouble when dealing with continuous-time stochastic processes". Which implies he is not keen on 'structuralist POV', although it doesn't rule out the possibility. – Yemon Choi – 2010-04-08T16:20:14.830

16Something that you may want to consider is the fact that probability spaces are not the essential objects in probability, for at least two reasons. First, it is very common to change the underlying probability space, as long as the distributions of the relevant random variables remain the same. This allows to consider new events along the way. As suggested in Neel's answer, this may have a categorical formulation. But worst than that is the fact that often (every time martingales appear, at least) you want to leave the space unchanged and vary the sigma-algebra. – Andrea Ferretti – 2010-04-09T16:29:34.603

12Indeed, one of the major differences between measure theory and probability theory (besides the perspective being completely different) is that in measure theory one fixes one sigma algebra, and in probability one considers relationships between multiple sigma algebras. – Mark Meckes – 2010-04-09T16:54:15.443

A paper from 2013: A categorical foundation for Bayesian probability by J. Culbertson and K. Sturtz

– Andrew MacFie – 2017-12-05T00:25:46.293