Timeline for Is there a maximal translation-invariant extension of Lebesgue measure?
Current License: CC BY-SA 4.0
8 events
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| Jul 18, 2023 at 21:31 | comment | added | Elliot Glazer | @VivaanDaga Sorry, I hadn't seen the "sigma" in your question. A finitely additive measure $\mu$ with those properties is maximal iff its domain is $\mathcal{P}(\mathbb{R})^d,$ so there is such an extension iff $d \le 2.$ Section 2 of the Ciesielski/Pelc paper constructs a partition $\mathbb{R}^d=\bigcup_{k<\omega} N_k$ such that for any $k$ and isometries $\langle g_m \rangle,$ there are uncountably many disjoint isometric copies of $\bigcup_{m<\omega} g_m ``(N_k).$ Fix $S \subset \mathbb{R}^d.$ By maximality of $\mu,$ we have $\mu(S \cap N_k)=0,$ so $S$ is in the domain of $\mu.$ | |
| Jul 9, 2023 at 5:51 | comment | added | Vivaan Daga | Or is that an open problem?(It feels like a hard question) | |
| Jul 8, 2023 at 20:03 | comment | added | Vivaan Daga | @ElliotGlazer We get an algebra by Zorn's lemma, not a sigma algebra unless I am missing something. | |
| Jul 8, 2023 at 19:53 | comment | added | Elliot Glazer | @VivaanDaga Yes, by Zorn's lemma. | |
| Jul 8, 2023 at 18:59 | comment | added | Vivaan Daga | Is there a finitely additive isomery invariant extension of Lebesgue measure(in $\mathbb{R}^d$) to a maximal sigma algebra? | |
| Jan 27, 2021 at 8:49 | comment | added | aduh | Cool paper, thanks! | |
| Jan 27, 2021 at 8:48 | vote | accept | aduh | ||
| Jan 27, 2021 at 8:30 | history | answered | Elliot Glazer | CC BY-SA 4.0 |