Talk:Equivalence of Definitions of Sigma-Algebra

Definition 2 is wrong?

 * Explain why. --prime mover (talk) 05:09, 30 May 2022 (UTC)

Counterexample
Let $X=\set {1,2,3,4}$ and $\Sigma = \set {\emptyset, X, \set {1,2}, \set{3,4}, \set {1,3}, \set {2,4}}$.

$\struct {X,\Sigma}$ satisfies Definition 2 but not Definition 1.


 * I believe the origin of this is misquotation. The source of the definition defines condition 2 to be $A \setminus B$ instead of $\map {\complement_X} A = X \setminus A$. I'm not sure of the ramifications of this adjustment so I'll leave my interference at that. &mdash; Lord_Farin (talk) 12:50, 30 May 2022 (UTC)


 * I see some of this stuff was posted up by me when I first tried to take on Avner Friedman but lost interest on page 3. I don't know where the unsourced presentation of the definition of sigma algebra comes from, haven't a clue where I got it from, but it's not from Friedman. Might have been off the web. I will take a look later and see what sense I can make of it. I may end up removing unsourced material if I cannot resolve its soundness. --prime mover (talk) 16:34, 30 May 2022 (UTC)


 * There is a second definition in Friedman as Theorem 1.1.1 which roughly corresponds to def 2. &mdash; Lord_Farin (talk) 17:11, 30 May 2022 (UTC)


 * Now I've got to revisiting Friedman, I find that the original Definition:Algebra of Sets page was written back in 2009 from a user no longer current. This definition was not associated with a source of any kind, and was presented in the form:

Given a set $X$ and a collection of subsets of $X$, $\mathcal{A} \subset \mathcal{P} \left({X}\right)$, $\mathcal{A}$ is called an algebra of sets if, given that $A, B \in \mathcal{A}$,

(1) $A \cup B \in \mathcal{A}$

(2) $c(A) \in \mathcal{A}$

where $c(A)$ is the complement of $A$


 * The construct Definition:Ring of Sets was never expressed in that form. It was always done in terms of $A \setminus B$ or $A \symdif B$, never $\map \complement A$.


 * So I recommend we "just" replace that definition (as has been established as flawed, and does not match the literature) with the "correct" one, that matches Friedman, who just says "a ring R is called an algebra if it has the additional property: $X \in R$" and take it from there. We will need to review any of the pages linking to this definition (and/or any specific subpages) and check they don't use the $\map \complement A$ version. If they do, they will need to be fixed.
 * I'll just get on with that then, and look at the refactoring of nests and chains after I've done that, then. --prime mover (talk) 19:31, 30 May 2022 (UTC)

Can someone else do it?
I see that the definition as given in Wikipedia uses the complement definition as matches the Definition 1 version. So we do have two apparently contradictory definitions of a sigma algebra.

Either Friedman is wrong or Wikipedia and all other sources (Cohn?) are wrong. Or both are wrong and everybody's wrong and we give up.

Someone else is going to have to resolve this. Someone clever, that is, someone not me. I would post it up on Math Stack Exchange but the latter has imploded under the weight of obnoxious gatekeeperism and is now utterly useless. --prime mover (talk) 19:55, 30 May 2022 (UTC)


 * The wikipedia definition I saw is exactly Definition 1. Both Lord_Farin's alternative with set difference and prime mover's alternative with algebra of sets seem equivalent to that. I have never seen contradictory definitions of a sigma algebra. The concept is mature enough. I could improve the page gradually but someone else may do it faster.--Usagiop (talk) 20:12, 30 May 2022 (UTC)


 * But you demonstrated that the definitions are not directly equivalent. That will have to be resolved. And I wouldn't know where to start on that. --prime mover (talk) 20:31, 30 May 2022 (UTC)


 * Yes, the current formulation of Definition 2 is wrong. As Lord_Farin also pointed out, something is missing. For example, if $\Sigma$ is not only a system of subsets of $X$, but an algebra of set of $X$, then (SA1')-(SA3') imply (SA1)-(SA3) of Definition 1.--Usagiop (talk) 20:44, 30 May 2022 (UTC)


 * Cohn only gives countable intersection and countable union as two definitions, it has nothing on disjointness. It points out that we can wlog replace "closed under countable union" with "closed under countable union of increasing sequences of sets" or "closed under countable intersection" with "closed under countable intersection of decreasing sequences of sets". Though I am not personally a fan of "overloading" definition pages, and should keep stuff like that to theorems. (it's given as a theorem with Cohn) I haven't noticed any errors in Cohn, I've found it to be a very reliable text. Caliburn (talk) 20:57, 30 May 2022 (UTC)


 * Yes, there are a lot of equivalent definitions. Without doubt, Definition 1 is the basic one. Although I actually don't know any importance of other equivalent definitions, it may be helpful to gather all such equivalent formulations here. --Usagiop (talk) 21:08, 30 May 2022 (UTC)