Timeline for There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions
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| Nov 30 at 15:19 | comment | added | Pierre PC | @PedroLauridsenRibeiro But frankly, I cannot do the exercise myself; what is clear is that the weak* topology and the weak cylindrical $\sigma$-algebra are generated by a common basis, but this only shows the direction "cylindrical implies weak*-Borel", and for the reverse direction I cannot seem to be able to find enough countability leverage since there is no first countability. Also I don't see how the argument applies to the strong topology (the weak topology here is the same as the weak* topology). | |
| Nov 30 at 15:07 | comment | added | Pierre PC | @PedroLauridsenRibeiro Thank you for your comment! I should have linked that question, that is where I learned about those two references... I am still a bit puzzled by all this. I think the closest result I could find in your reference is the the beginning of section IV-2.4, p330, where it is left as an exercise to show that, in my terminology, the Borel $\sigma$-algebra of the weak* topology coincides with the weak cylindrical $\sigma$-algebra (to be continued). | |
| Nov 29 at 19:11 | comment | added | Pedro Lauridsen Ribeiro | You may want to check out mathoverflow.net/a/157219/11211 and the discussion in the comments... TL&DR: Gel'fand-Vilenkin's book, "Generalized Functions - Volume IV", Chapter IV, pp. 303ff.. | |
| Nov 29 at 16:43 | history | asked | Pierre PC | CC BY-SA 4.0 |