Timeline for Examples of metric spaces with measurable midpoints
Current License: CC BY-SA 4.0
9 events
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| Jun 30, 2020 at 12:57 | comment | added | dohmatob | Great. Thanks for the clarification. So, it might be possible to get the existence of a Borel measurable function with requirements less than Polish, right ? | |
| Jun 30, 2020 at 12:54 | comment | added | Arno | Yes, every Baire class 1 function is Borel measurable. The set of Borel measurable functions is the union of the Baire class $\alpha$ functions where $\alpha$ ranges over the countable ordinals. Hence, the "much simpler" statement. | |
| Jun 30, 2020 at 12:49 | comment | added | dohmatob | I don't understand what you mean by "Baire is simpler than Borel measurable". When all the dust settles, do we have a measurable function ? (Note that I don't care about computation, construction, or simplicity; only existence) | |
| Jun 30, 2020 at 12:35 | comment | added | Arno | I've clarified that Baire class 1 is much simpler than Borel measurable. The Polish-requirement is what makes sure that Baire space occurs as subscript where it does. | |
| Jun 30, 2020 at 12:34 | history | edited | Arno | CC BY-SA 4.0 |
added 117 characters in body
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| Jun 26, 2020 at 9:11 | comment | added | dohmatob | Indeed, this is done in Theorem 4.1 of cca-net.de/vasco/publications/borel.pdf. | |
| Jun 26, 2020 at 8:59 | comment | added | dohmatob | Also, my 2 cents is that a version of the arguments you use here can be in principle used to prove the Kuratowski-Ryll-Nardzewski measurable selection theorem used in Gerald Edgar's answer. | |
| Jun 26, 2020 at 5:32 | comment | added | dohmatob | Thanks. Upvoted. There are many things which seem to be missing. You seem to be using the fact that Polishness of the space implies many things, but it's not clear by what mechanism these conclusions are drawn. Because we're in a metric space, the assumption in case 1 is equivalent to saying that $X$ is locally compact (at this point, I'm fine with such an assumption, since the other posted solutions require it too). Finally, is every 'Baire class 1" function also Borel-measurable ? | |
| Jun 25, 2020 at 21:26 | history | answered | Arno | CC BY-SA 4.0 |