Wikipedia:Reference desk/Archives/Mathematics/2022 September 13

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September 13

Axiom of choice, and probability theory

Is AC relevant much to probability theory? Particularly, we're taught that a proability space is a sample set plus an event σ-algebra, but we use the machinery of σ-algebras instead of just saying that the event set is all the subsets (i.e. the power set) of the sample set, to get around the possibility of badly behaved subsets like in the Banach-Tarski paradox. But those subsets only arise in the presence of AC or some weaker version of AC. There is a famous theorem that models of ZF (without AC) exist in which all sets of reals are Lebesgue measureable.

So my question is: does AC figure into significant theorems of probability? Is it "morally" ok to think of the event set as all the subsets of the sample space and not worry too much about proofs failing due to that? I mean just for the purpose of figuring stuff out. To write real proofs of course you need to account for all the machinery. Thanks. 2601:648:8201:5DD0:0:0:0:256B (talk) 00:47, 13 September 2022 (UTC)

In certain logics one can appeal to AC to prove the existence of bad boys like in the Banach–Tarski paradox. Do they cease to exist when one weakens the logic, or are they still there, only one cannot now prove their existence? The point of the restriction is that the generalization to infinite probability spaces needs the events to be measurable. When the sample space is a subset of Rⁿ, the concepts of Lebesgue integral and Lebesgue measure are convenient, which leads in a natural way to σ-algebras (see Carathéodory's criterion). But σ-algebras are more general; they do not need to have a topological interpretation, so one can prove results more abstractly and more generally. Does the restriction of the event space to a σ-algebra make any proof more complicated? I'd expect that they actually simplify proofs, since one doesn't have to worry about running into a paradox.  --Lambiam 09:06, 13 September 2022 (UTC)

Thanks. The famous theorem I mentioned is the Solovay model, which assumes ZF with the negation of AC (since AC proves the Banach-Tarski paradox). In the Solovay model, there are no non-measurable sets. What I wondered was whether any of the standard theorems of probability somehow had uses of AC lurking within. You are probably right that the restriction of the event space to a σ-algebra doesn't really complicate proofs. It does, however, make the concept of a probability space a little bit harder to explain. 2601:648:8201:5DD0:0:0:0:256B (talk) 06:24, 14 September 2022 (UTC)

In my view, the best way to think of it is the axiom of choice is "really" true (if you're a Platonist like me you can leave off the scare quotes, but if not go ahead and leave them on). Leaving aside "reality", it's just the most natural way to think of sets, once you've understood the mental picture of sets corresponding to the von Neumann universe (sets built up from nothing by transfinitely iterating the powerset operation).

However, for purposes of probability and so on, you can remember that the nice properties you'd like to have, such as measurability and so on, apply to all "reasonably definable" sets, including practically any particular named set you're likely to see show up in a proof, unless you're explicitly choosing representatives of equivalence classes and so on.

To get a little more precise, it follows from ZF plus the axiom of determinacy (which is incompatible with the axiom of choice) that all sets of reals are Lebesgue measurable and have various other nice regularity properties (property of Baire, perfect set property). Generally the sets you're looking at in probability theory can be thought of as sets of reals.

How do you reconcile the two views, since AC and AD contradict one another? Best way to think of it is that AC is true in the complete "real" world, but AD is true in a restricted inner model that contains all the "real" real numbers, but leaves out some sets of reals, including all the "pathological" ones.

Specifically, there's a model called L(R) that satisfies AD (assuming some large cardinals), that contains all the "real" reals, and that contains pretty much any "explicit" set of reals you're likely to think of, unless you're doing something very unusual for probability theory.

So you can keep the naturalness of AC in your mental model, but still be comfortable that any particular set you come across is going to be in L(R), and therefore have all the nice regularity properties. --Trovatore (talk) 17:15, 17 September 2022 (UTC)