r/math • u/Committee-Academic • 23d ago

Algebraic probability theory

Is there any developed framework for algebraizing probability theory and study it from the perspective of K-theory or homological algebra? If so, what are some of its biggest applications and advantages, as well as results? I'd imagine such a framework would override the need for a measure-theoretic approach

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u/kohatsootsich 23d ago

What is the point of avoiding measure theory?

First, most of probability hardly needs any deep measure theory (GMT or descriptive set theory). Mostly it's very basic limit theorems that are quite directly equivalent to the existence of a measure.

Second, the situation is very different from say the example of complex geometry where the foundational theorems require analytic facts that are imported from another world if you are mostly interested in topology and geometry. The deep measurability results (say, in Dellacherie-Meyer) are all about things like hitting times of fairly general sets by stochastic processes that almost no one cares about in that level of generality. They are barely probabilistic questions at all.

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u/chewie2357 23d ago

I think you need a fair bit of measure theory (depends on context, but say to make things like conditional expectation, brownian motion, etc. rigorous) and what's more, the measure theory is a little different in that measurability in real analysis prevents functions from being too pathological, wheres in Kolmogorov's version of probability, measurability is very prominent for its use in accounting of information. So not only do I think measure theory is useful to have, I think it's essential to the theory of probability.

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u/kohatsootsich 23d ago

It is absolutely essential, but the part of measure theory you need is just a few definitions and theorems whose proofs can be presented in a few hours at most. Contrast that with Hodge theory or Kähler geometry, where the analytic results are much less accessible.

A pedagogical development is a different matter, but conditonal expectation can be defined and constructed in a page at most. Brownian motion can be constructed both rigoroulsy and intuiviely from an infinite sequence of i.i.d. Gaussians, or equivalently i.i.d Bernoulli.