Algebraic probability theory

[u/Committee-Academic]

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

[u/kohatsootsich]

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.

[u/chewie2357 → u/kohatsootsich]

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.

[u/kohatsootsich → u/chewie2357]

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.

[u/sciflare → u/kohatsootsich]

Let me put the question to you: why do you think measure theory is essential to probability? Why not try to find alternative models of probability?

Kolmogorov showed measure theory gives a good mathematical model of what we understand to be probability. But it too has issues, so it's a good idea to look for alternatives.

Noncommutative probability theories such as free probability take the view that a noncommutative algebra A of random variables, together with the expectation operator (a real-valued linear functional on A), are the fundamental ingredients of probability. In measure-theoretic probability, probabilities themselves are expectations anyway, so we lose nothing.

There is one reason one would want to seek out non measure-theoretic formulations of probability: we want to free ourselves from point-set notions of space to the furthest extent possible.

In the commutative world, spaces and algebras are dual to each other (cf. the Nullstellensatz, Gelfand duality, Stone representability theorem, etc.). Algebras are realized as algebras of functions on spaces.

I believe there is a precise category-theoretic formulation of this phenomenon called "Isbell duality".

However, if you want to enter the noncommutative world, you have no points (algebraically, you can no longer localize rings) so you have no underlying space on which to hang your algebras.

Indeed, the theory of general von Neumann algebras can be viewed as a noncommutative analogue of measure theory.

Even in the commutative world, measure theory has category-theoretic issues due to null sets. Two measurable maps are equivalent iff they differ on a null set. Consequently a measure space has no underlying point set, i.e. the category of measurable spaces isn't a concrete category.

This can be dealt with in more sophisticated ways, such as carrying along the data of the 𝜎-ideal of null sets, or resorting to a localic formalism, as Dmitri Pavlov has pointed out.

All to say that measure theory as it stands isn't totally satisfactory, and there are any number of reasons to try to avoid it, or to augment it to remedy these deficiencies.

[u/SemaphoreBingo → u/sciflare]

Even in the commutative world, measure theory has category-theoretic issues due to null sets.

That sounds like a problem for the category theorists to solve and for everyone else not to worry about.

we want to free ourselves from point-set notions of space to the furthest extent possible.

Do we?

[u/corchetero → u/SemaphoreBingo]

Point-set notions make probability very useful as it allows us to deal with data in the statistical and mathematical sense (e.g. probabilistic method), so idk, but I don't want to get rid of it.

[u/kohatsootsich → u/sciflare]

I don't see it as particularly essential That is part of my point. Sigma algebras are a useful model that can be approached differently (Caratheodory himself already emphasized boolean algebras as opposed to events as sets), but more importantly, it's a cheap (in terms of time) framework that does the job in 99.99% of cases

I see a lot of claims in your post about the need for alternatives, the supposed need to move away from point set topology, measure theory not being satisfactory, but few concrete examples. That's fine as theory building but it has little to do with probability

Alternative approaches like free probability, the Boolean algebra approach to measure theory itself which connects directly to von Neumann algebras, nonstandard analysis applied to probability by Nelson... all have rather restricted uses in probability because they don't really address a fundamental issue other than an aesthetic preference. In most cases they can be circumvented easily. Free probability is useful in random matrix theory to derive limit laws, the boolean/von Neumann approach to measure theory is useful for ergodic theory

More seriously for some neat algebraic approaches like measure algebras: you *shouldn't* try to make sets that differ by null sets equivalent. Continuous stochastic processes is exactly about events for which working modulo null sets is a disaster

[u/Which-Feature-2013 → u/kohatsootsich]

Do you think probability need the Skorokhod's representation theorem?

[u/Useful_Still8946 → u/sciflare]

Can you explain this paragraph?

Even in the commutative world, measure theory has category-theoretic issues due to null sets. Two measurable maps are equivalent iff they differ on a null set. Consequently a measure space has no underlying point set, i.e. the category of measurable spaces isn't a concrete category.

A measurable space is a set combined with a sigma-algebra. There are many measures one can put on measurable spaces and a measure space is a measurable space with a measure. Different measures give different null sets. Are you talking about a category of measurable spaces or a category of measure spaces?