Timeline for $L^1_{\mu}$ as limit
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Jan 20, 2020 at 10:16 | history | bounty ended | ABIM | ||
| S Jan 20, 2020 at 10:16 | history | notice removed | ABIM | ||
| S Jan 14, 2020 at 8:19 | history | bounty started | ABIM | ||
| S Jan 14, 2020 at 8:19 | history | notice added | ABIM | Reward existing answer | |
| Jan 13, 2020 at 14:06 | comment | added | ABIM | Perfect! I'll work this out then; thanks. | |
| Jan 13, 2020 at 14:02 | comment | added | user131781 | Well, no but it is so elementary that I don‘t think that one needs a reference. It is simply the fact that if you have a compatible, bounded sequence $(f_n)$ where $f_n$ is in $L^1([-n,n])$, then they combine to form an integrable function on the line. The only possible finesse comes from the fact that the functions are only defined a.e. but the countability condition takes care of that. Of course, I only chose the real line as an example—-this works for any $\sigma$-finite meaure. A useful toy example to make everything transparent would be the positive integers with counting measure. | |
| Jan 13, 2020 at 13:39 | history | edited | ABIM | CC BY-SA 4.0 |
added 109 characters in body
|
| Jan 13, 2020 at 10:42 | comment | added | ABIM | @user131781 would you happen to have a reference on $L^1$ described this way in the Category of Banach spaces and (linear?) contractions? | |
| Jan 12, 2020 at 5:51 | comment | added | user131781 | I can‘t think of one for this explicit situation but both follow directly from the general construction of such projective limits as compatible threads, in this case as compatible sequences $(f_n)$ with $f_n \in L^1(\mu_n)$. In the lcs case there is no growth condition so they combine to form a locally integrable function, in the Banach space one, the $L^1$ norms are bounded so you get a globally $L^1$ function. | |
| Jan 11, 2020 at 22:38 | comment | added | ABIM | This is very interesting actually. Could you possibly provide a reference to this; maybe some book? | |
| Jan 11, 2020 at 16:53 | comment | added | user131781 | Taking projective limits of Banach spaces in the sense of lcs’s will not usually produce a Banach space, as pointed out below. However, the category of Banach spaces with contractions as morphisms does have projective limits and I would suggest that this might be what you want. As a simple example, if $\mu$ is Lebesgue measure on the line, and $\mu_n$ is its restriction to $[-n,n]$, the sequence $(L^1(\mu_m))$ has the corresponding $L^1$-space as projective limit in the second sense, but $L^1_{loc}$, the Fréchet space of locally integrable functions, in the first. | |
| Jan 11, 2020 at 13:12 | vote | accept | ABIM | ||
| Jan 11, 2020 at 13:12 | answer | added | Jochen Wengenroth | timeline score: 7 | |
| Jan 11, 2020 at 11:12 | history | asked | ABIM | CC BY-SA 4.0 |