# 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