Definition talk:Lebesgue Space

In light of the fact that existing references to the Lebesgue space refer to $\mathcal L^p$ and not the quotient $L^p = \mathcal L^p/\sim$, I will be going forward with "Lebesgue Space" for $\mathcal L^p$ and will call $L^p$ "Lp spaces", with Definition:Lp Space etc., unless there are objections. I've put off actually getting this done long enough. Caliburn (talk) 17:52, 18 May 2022 (UTC)

Draft - to do: prove that A.E. equality is an equivalence relation, define the norms, define the integrals of equivalence classes and show that these are well-defined (this is basically already on here in some form), define pointwise addition/pointwise multiplication of functions and show that these are also well-defined. Caliburn (talk) 18:01, 18 May 2022 (UTC)

Good plan. The devil is always in the details. --prime mover (talk) 18:19, 18 May 2022 (UTC)

I think that this is fine. Some points of attention:

• A.E. equality is not how the current quotient is defined (of course it is equivalent) so it is not immediate that the definition is sensible. I would prefer to stick with the "seminorm of difference is $0$" and then prove that this is a.e. equality.
• Please include something like {{about}} on both pages referring to the other one.
• Please include references/disambiguation to the Definition:P-Sequence Space $\ell^p$. In particular also with regard to Definition:P-Norm and Definition:P-Seminorm. Thinking out loud, maybe "of Function" and "of Sequence" are suitable suffixes.
• I vaguely recall that a lot of pointwise stuff is already defined, so you should be able to save yourself some work there. E.g. Pointwise Addition preserves A.E. Equality

Thanks for your work in this area. — Lord_Farin (talk) 19:19, 18 May 2022 (UTC)

Well I've quickly rediscovered why I've been putting it off so long, I keep on tripping over myself and starting over. I will probably start over in the morning. The temptation is to preserve $\norm {\eqclass f \sim}_p = \norm f_p$, but I do want to define the integral of an equivalence class. Whether to just define this as the $1$-norm of a representative (probably invoking Choice somehow) or have a bespoke definition and relate it back to the norm, I'm not sure and I'm tossing between the two. I had noticed the first thing but it must have slipped my mind, I'll change my draft to that. And I did plan on making clear that $\ell^p$ is just an $L^p$ space on $\struct {\N, \mathcal P \N}$ with the counting measure as a theorem. The existing A.E. equality pages were just things I did on the side when I was proving something else. It shows something slightly different anyway - that you can add or subtract stuff in A.E. equality like the usual $=$, it doesn't show that if $f = g$ ae and $a = b$ ae that $a + f = b + g$, which is what you need for addition to be well-defined. (the first might imply the second with a clever choice of $h$, can't think what though) Caliburn (talk) 19:47, 18 May 2022 (UTC)

Admittedly nitpicking, but the $1$-norm is not the integral. It's the integral of the absolute value. I'm afraid there is no easy way around defining all the derived concepts with painstaking attention and proofs regarding the a.e. situation. Probably the most elegant inroad is to prove that the space of a.e. classes is a ring and work from there. — Lord_Farin (talk) 07:06, 19 May 2022 (UTC)

Yeah overlooked that. Somewhat of a compromise: maybe having a definition page for $\sim$ and having both the seminorm and a.e. equality versions as two definitions then an equivalence proof? Caliburn (talk) 08:29, 19 May 2022 (UTC)

That is an elegant and simple solution. Nice! — Lord_Farin (talk) 08:32, 19 May 2022 (UTC)

User:Caliburn/s/mt/Definition:Lp Space is basically final now. It goes through some trouble of distinguishing three objects: the underlying set $L^p$, the vector space $L^p$, and the normed vector space $L^p$. I hope it's clear that this is what's being done. Caliburn (talk) 10:05, 19 May 2022 (UTC)

Just putting here a note that I haven't forgotten about this progress, but I am currently otherwise occupied. I expect to be able to go through your work somewhere this weekend. — Lord_Farin (talk) 08:32, 20 May 2022 (UTC)

Fair enough, I'm making life increasingly difficult for myself because I realised I've not emphasised the dependence of $\sim$ on $\mu$ explicitly. I probably won't bother doing this past the very basic definitions, but I feel I should close-to-metal. Caliburn (talk) 12:01, 20 May 2022 (UTC)