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. &mdash; 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)