# Pointfree probability theory

I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like pointfree topology, where one basically replaces topological spaces by their locales of open sets, I figured there is a way to do something similar with $\sigma$-algebras and with probability spaces.

Any thoughts on that? Does somebody know, whether this has been studied before?

Here are some more thoughts: I suppose a problem is how to recover the sample space $\Omega$ from a pointfree probability space, as there is a no guarantee that there is an injection $\Omega \to \sigma$ from the sample space to the $\sigma$-algebra of a probability space. I wonder, how important it is to have a sample space at all. I (think I) know, that probability theory is actually about random variables, but do we really need a sample space to talk about those? Also, considering that there is no obvious notion of a morphism between probability spaces, maybe there are other objects we should look at?

• I would consider asking this on MO. – zhoraster Aug 29 '15 at 9:53
• @zhoraster I don't classify as "professional mathematician" unfortunately. So should I really do this? – Stefan Perko Aug 29 '15 at 9:56
• @ Stefan Perko, not a professional? You speak about categories, morphisms, $\sigma$-algebras, and I guess these are not just fancy words for you. Yes, I think this question is good for MO. – zhoraster Aug 29 '15 at 10:07
• mathoverflow.net/questions/20740/… Not completely the same, but relevant points regarding the abstract/categorical approach to probability theory. – Forgottenscience Aug 29 '15 at 10:17
• Now posted also on MathOverflow: A Point-free probability theory? – Martin Sleziak Jul 24 '17 at 12:36

## 1 Answer

Indeed, by the Kolmogorov extension theorem any systems of random variables (stochastic processes) can be represented by a suitable set of finite dimensional cumulative distribution functions.

However, handling (just to mention one important example) the so called tail events, that is, events relating to the limiting behavior of infinite series of random variables, would be incredibly hard without assuming the existence of an underlying probability space.

Reading the above mentioned Wikipedia article is recommended.

• Dear @StefanPerko, I am running out of time for now. But I promise you that I will come back in the evening. As far I can guess we are in the same time zone... – zoli Aug 29 '15 at 11:31