Talk:Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined

I have no attachment to the term "quotiented", but I'm not sure I'm a fan of "Space of A.E. Equal Measurable Functions". To me this means a set of functions which all agree A.E., when we're actually looking at a set of such things. Would we be happy with "identified by A.E. Equality"?. I can add an expansion of the definition of $\map {\mathcal L^p} {X, \Sigma, \mu}/\sim$ to the $L^p$ page. Caliburn (talk) 17:35, 19 May 2022 (UTC)

Also I think I will need to give $\map {\mathcal M} {X, \Sigma, \R}/\sim$ its own page when I show that it is a vector space. Caliburn (talk) 17:36, 19 May 2022 (UTC)

Well, formally speaking that'd be correct, but my idea was that it is clear that such a space is uninteresting so that (equivalence classes of) would be implicit. "Identified by AE Equality" is indeed less bad than "quotiented". Even "Quotient Space of MF by AE Equality" would already be more bearable. I'm trying here to retain a sane page name length that doesn't make you lose track halfway through.

Incidentally as I see "quotient space" I am realising that you can probably use some of the nice results that were recently posted in the field of Definition:Relation Compatible with Operation i.e. Definition:Congruence Relation to save you some work in proving things well-defined. — Lord_Farin (talk) 17:58, 19 May 2022 (UTC)

I've moved everything to "identified by" for now. I don't think the page name is overly long compared to some others we have, eg. Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval. I've put a clearer outline of the pages that are being worked on here: User:Caliburn/s/mt/LpStuff as well. My intent was to prove everything very directly, and if there are any clever tricks or appeals to more general theory those can come in as second proofs. Caliburn (talk) 08:23, 20 May 2022 (UTC)