Wikipedia:Reference desk/Archives/Mathematics/2022 November 28

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November 28

sigma algebras of null sets

Per https://ncatlab.org/nlab/show/probability+theory

Some argue that in the study of measure and probability, one should start not only with sigma algebra of measurable sets but also another of null sets. Somehow this is abstractly captured by the approach of commutative von Neumann algebras.

What do they mean by that? In classical probability, null sets are events that almost never happen, and the same is true of their countable unions, right? So that is a set of events of probabillity 0. Why is that interesting? Would it make more sense if I studied enough functional analysis to understand what a von Neumann algebra is? At the moment, I can't make any sense of the article about them. Thanks. 2601:648:8200:990:0:0:0:497F (talk) 06:27, 28 November 2022 (UTC)

It is unclear (to me), and the reference to commutative von Neumann algebras does not help to make things any clearer. Imagine the σ-algebra $(X,\Sigma )$ of a measure space and a somehow related σ-algebra $(Y,\Sigma ')$ of null sets. Since σ-algebras are closed under complementation and finite union, not only would any element $S\in \Sigma '$ have measure 0, but so would its complement $Y\setminus {S}$ and therefore $Y$ itself. What would be the significance of $Y$ in relation to $X$? Unfortunately, the cited text gives no clue about the supposed benefit, let alone the argument. If we knew who the "some" are who argue this, studying their arguments might clarify the issue, but in this regard we are also left in the dark.  --Lambiam 13:33, 28 November 2022 (UTC)

Thanks, maybe I can somehow ask over there. 2601:648:8200:990:0:0:0:497F (talk) 22:23, 29 November 2022 (UTC)

Given the categorical slant of ncatlab, this MathOverflow question might be relevant? Felix QW (talk) 18:00, 4 December 2022 (UTC)

I've not tried to understand the top-rated answer to the MathOverflow question or even read it through to the end, but I noticed that the added structure of "negligible sets" is not just a σ-algebra (as in the quote from the article on ncatlab) but a σ-ideal, presumably (this is not stated) a σ-ideal of the σ-algebra of measurable sets already referred to. I also suppose (this is also not stated) that this σ-algebra of measurable sets is equipped with a measure, making it a measure space, and that this is the measure by which the "negligible sets" have measure 0.  --Lambiam 01:06, 5 December 2022 (UTC)

Absolutely. But as you rightly pointed out, it makes no sense for the negligible sets to be closed under complement, so I assumed that what the ncatlab wanted to write is $\sigma$-ideal instead of $\sigma$-algebra.

Regarding the second point, I think the whole idea of adding some formalisation specifically of null sets is to obtain a structure richer than a measurable space, but weaker than a measure space. So if you are also given a measure, then the whole construction is moot because you get null sets from the measure rather than adding them to your structure. Felix QW (talk) 11:21, 5 December 2022 (UTC)

Good luck in doing probability theory, categorically or otherwise, without a measure of the non-negligible events.  --Lambiam 14:45, 5 December 2022 (UTC)