Timeline for Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC
Current License: CC BY-SA 3.0
10 events
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| Oct 15, 2013 at 22:44 | comment | added | Asaf Karagila♦ | @Nate: I checked the papers, and Wright actually used the Baire property; and Garnir's paper is very unclear to me as to what and where is the proof there. But it seems to me that he really places a lot of focus on the Lebesgue measurability. If I recall correctly some homomorphisms satisfy automatic continuity from measurability (e.g. $f\colon\Bbb{R\to R}$ as a $\Bbb Q$-vector space homomorphism) and so I would expect the proofs not to be very different, but I can't come up with an argument which doesn't involve sleeping right now. | |
| Oct 15, 2013 at 22:41 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
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| Oct 15, 2013 at 15:31 | comment | added | Asaf Karagila♦ | @Nate: In Solovay's model every set of reals has the Baire property as well. But one can prove that directly from measurability in a similar manner. That's what Wright and Garnir essentially do, if I recall correctly. I'll put some clarifications later on tonight (there was a minor addition I wanted to make anyway). Thanks for the comment. | |
| Oct 15, 2013 at 15:29 | comment | added | Nate Eldredge | I can see why Pettis's theorem implies automatic continuity in Shelah's model (if every set is Baire measurable, then every homomorphism is Baire measurable hence continuous). What do you do in Solovay's model? It's not clear to me how to exploit Lebesgue measurability. | |
| Oct 14, 2013 at 18:27 | comment | added | Asaf Karagila♦ | I hope you will find the references, some are trickier to find than others. | |
| Oct 14, 2013 at 18:27 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
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| Oct 14, 2013 at 18:12 | comment | added | Asaf Karagila♦ | I was actually trying to come up with a good reference for those. Solovay's construction is "rather simple" (as a construction of a symmetric extension), but Shelah's model is much much more complicated. | |
| Oct 14, 2013 at 18:05 | comment | added | mathcounterexamples.net | Thanks Asaf for your answer. Maybe you have a reference for the construction of such models? | |
| Oct 14, 2013 at 18:04 | vote | accept | mathcounterexamples.net | ||
| Oct 14, 2013 at 17:55 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |