Timeline for Proof of the Dunford-Pettis theorem in the context of probability spaces
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 20, 2023 at 20:56 | comment | added | Iosif Pinelis | @DirkWerner : Yes, I should have said that this does complete the proof. | |
| Nov 20, 2023 at 19:22 | comment | added | Dirk Werner | Well, this does complete the proof since the metrisability of weakly compact sets in separable spaces is a well-known fact (showing the easy half of Eberlein-Shmulyan). | |
| Nov 20, 2023 at 17:56 | comment | added | Iosif Pinelis | @DirkWerner : Yes, this would complete the proof. | |
| Nov 20, 2023 at 17:50 | comment | added | Dirk Werner | One needs the additional information that weakly compact sets in separable Banach spaces (like the closed linear span of the $X_n$) are metrisable; therefore one actually can extract a weakly convergent subsequence. | |
| Nov 20, 2023 at 17:12 | comment | added | Iosif Pinelis | @JohnDawkins : It seems that the problem with this approach is that the implication (1)$\implies$(2) in Theorem 25-II of Dellacherie--Meyer -- approximately corresponding to the implication 2$\implies$1 in the OP -- would only give a weakly convergent subnet, rather than a subsequence, of the sequence $(X_n)$. | |
| Nov 20, 2023 at 0:53 | comment | added | John Dawkins | But the give a short measure-theoretic proof of a version of E-S that is sufficient for their purposes. | |
| Nov 19, 2023 at 18:47 | comment | added | rfloc | Thank you for your answer! Unfortunately the book also uses the Eberlein–Smulian Theorem to prove that equivalence! | |
| Nov 19, 2023 at 18:07 | history | answered | John Dawkins | CC BY-SA 4.0 |