Timeline for Spaces with atomless independent $\sigma$-sub-algebras
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 27, 2022 at 9:02 | vote | accept | Thomas Anton | ||
| Mar 26, 2022 at 6:54 | comment | added | Michael Greinecker | @RabeeTourky I hope someone can resolve the second question before any answer is accepted. | |
| Mar 26, 2022 at 1:38 | comment | added | Rabee Tourky | Should the poster accept your answer? Because it is correct for first question, but second is difficult but seems interesting. | |
| Mar 25, 2022 at 19:32 | comment | added | Michael Greinecker | @RabeeTourky I've started to become agnostic to what the answer to the second question is. | |
| Mar 25, 2022 at 19:11 | comment | added | Rabee Tourky | Yes that's right, take one copy of [0,1] in there space and construct the example X mathoverflow.net/questions/54033/… the rest of the copies of [0,1] will U be independent of X and the completion of U contains Sigma. In these spaces countably generated subalgebras don't matter for the measure algebra. | |
| Mar 25, 2022 at 19:00 | comment | added | Rabee Tourky | Yes I know. The question is closely related. Take a maximal U independent of X. Must the completion of U contain Sigma? it would be odd intuitively, and that is equivalent to my question. So if your conjecture is correct. The second question becomes, there is U independent of X whose completion contains all information Sigma | |
| Mar 25, 2022 at 18:54 | comment | added | Michael Greinecker | @RabeeTourky The question asks only about the case with $\mathcal{X}$ being countably generated. | |
| Mar 25, 2022 at 18:07 | comment | added | Rabee Tourky | Regarding the second question, there are complete strict Sigma-sub-algebra (not countably generated) that do not have independent none trivial events. Or is the claim that there are none? | |
| Mar 25, 2022 at 14:39 | comment | added | Michael Greinecker | @NateEldredge In the example, the problem is that there are not enough independent events. On the level of measure algebras, the second question has a positive answer. This follows from results on factor spaces in volume 3 of Fremlin's magnus opus on measure theory. If there are problems, they should have to do with managing null sets. | |
| Mar 25, 2022 at 14:21 | comment | added | Nate Eldredge | Nice example. My intuition is that the second part is probably false, and I am wondering if one of the examples from mathoverflow.net/questions/54033/… could be adapted. | |
| Mar 25, 2022 at 14:06 | history | answered | Michael Greinecker | CC BY-SA 4.0 |