Timeline for $X$ Polish geodesic implies $(P_2(X), W_2)$ geodesic
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 20, 2023 at 12:24 | answer | added | Mizar | timeline score: 0 | |
| Oct 21, 2013 at 8:17 | review | Close votes | |||
| Oct 21, 2013 at 13:07 | |||||
| Oct 20, 2013 at 18:53 | comment | added | Otis Chodosh | @User11111, I have posted a more detailed explanation. I think you were trying to apply measurable selection in the wrong direction. Hope this clears up some confusion! | |
| Oct 20, 2013 at 18:52 | answer | added | Otis Chodosh | timeline score: 3 | |
| Oct 20, 2013 at 18:08 | comment | added | User11111 | @TapioRajala Yes indeed, I'm sorry for the confusion. | |
| Oct 20, 2013 at 17:46 | comment | added | Tapio Rajala | If your previous question was aimed at proving the result stated here, it was the wrong question. Did you mean to ask it for the specific mapping $S$ giving for pairs $(x,y)$ all the geodesics from $x$ to $y$? | |
| Oct 20, 2013 at 16:58 | comment | added | User11111 | @OtisChodosh Thank you! What I would like to do is to apply Kuratowski Ryll-Nardzewski, which guarantees that I can extract a measurable selection from a multifunction $S$ as defined in mathoverflow.net/questions/145338/… if (among other conditions) the multifunction is measurable, i.e. if I have an open set $U$ in $\text{Geod(X)}$ then $S^{-1}(\lbrace U \rbrace)$ is Borel in the domain. I guess I'm getting something wrong, but can't figure out what... | |
| Oct 20, 2013 at 16:10 | comment | added | Otis Chodosh | @User11111, it seems you're confusing the statements: "there exists a measurable selection" with "every selection is measurable" . One place to read about measurable selection is Villani, "Optimal Transport: Old and New" p 92, and Corollary 5.22 | |
| Oct 20, 2013 at 15:41 | comment | added | User11111 | For example take a multifunction $S: X^2 \rightarrow 2^{\text{Geod}(X)}$. As @TapioRajala pointed out in mathoverflow.net/questions/145338/… it is not Borel-measurable. How can I expect to find a Borel measurable selection? | |
| Oct 20, 2013 at 14:28 | comment | added | User11111 | The part which causes me most trouble is actually missing: which selection theorem they use to obtain a Borel map Geodsel | |
| Oct 20, 2013 at 14:23 | comment | added | Tapio Rajala | See for instance Theorem 2.10 in [ Ambrosio & Gigli, A user's guide to optimal transport cvgmt.sns.it/media/doc/paper/195/users_guide-final.pdf ] | |
| Oct 20, 2013 at 13:46 | history | asked | User11111 | CC BY-SA 3.0 |