Timeline for Do you know important theorems that remain unknown?
Current License: CC BY-SA 4.0
15 events
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| Mar 6, 2023 at 10:26 | comment | added | MikeTeX | @Daniele tempieri. Isn't something missing in your condition: the fact that the sets $A_n$ are pairwise disjoint does not imply what it is supposed, unless the whole space has finite measure. For example, if we take $\mathbf R_+$ and the $A_n = [n, n+1[$, then these sets are pairwise disjoint but their Lebesgue measure is equal to 1. | |
| Dec 31, 2021 at 17:15 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Added direct link to Cody's answer as per suggestion of Spice (see comments)
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| Dec 31, 2021 at 17:14 | comment | added | Daniele Tampieri | @LSpice, thanks. I'll put the ink in the answer. | |
| Dec 31, 2021 at 17:12 | comment | added | LSpice | The answer of @coudy referenced here. | |
| Dec 31, 2021 at 17:07 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor addition (hyperlinked MR and Zbl reviews)
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| Oct 23, 2021 at 9:15 | comment | added | dohmatob | @DanieleTampieri Thanks for the reply, and more broadly, for Cafiero's theorem. | |
| Oct 22, 2021 at 12:27 | comment | added | Daniele Tampieri | @dohmatob no: we are really taking the full limit. The condition means that for any family $\{A_n\}_n$ such that the set theoretic limit is the empty set $\emptyset$, the numerical set function $\phi$ goes likewise to $0$ (you surely noticed that $\Cap_{m\ge j\ge n} A_j=\emptyset$ for all $m$). (P.S. I apologize for my later in answer to your comment, and good luck for your researches). | |
| Oct 21, 2021 at 14:29 | comment | added | dohmatob | @DanieleTampieri I'm struggling to get my head around the condition $\lim_n \phi(A_n)=0$ for all disjoint $(A_n)_n$, or did you mean $\liminf_n \phi(A_n)$ ? | |
| Jan 15, 2020 at 18:46 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Typo in page numbering
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| Jan 15, 2020 at 18:41 | comment | added | Daniele Tampieri | Hi @mathworker21. It is a clarification, perhaps a little pedantic: the definition of uniformly exhaustive I reported from [4] (Definition 1.1, p. 110) refers a simple set $H$, while in Cafiero's theorem we have a net $(f_n\cdot\mu_n)_{n\ge1} which is a particular kind of family. I pointed out that the definition includes that case, since a net is a special case of a (set theoretical) family and each set is a family with itself taken as the index (Halmos docet). Finally, apologies for the later in my answer to your comment: I am very busy now. | |
| Jan 14, 2020 at 13:59 | comment | added | mathworker21 | what does "(and thus possibly a family)" mean? | |
| Apr 3, 2018 at 9:56 | history | edited | j.c. | CC BY-SA 3.0 |
minor grammar fixes
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| Apr 2, 2018 at 19:17 | history | edited | Daniele Tampieri | CC BY-SA 3.0 |
Stated where I learned it.
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| S Apr 2, 2018 at 19:00 | history | answered | Daniele Tampieri | CC BY-SA 3.0 | |
| S Apr 2, 2018 at 19:00 | history | made wiki | Post Made Community Wiki by Daniele Tampieri |