Timeline for Measure theory in nuclear spaces
Current License: CC BY-SA 3.0
4 events
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| Dec 30, 2017 at 23:28 | comment | added | Abdelmalek Abdesselam | I agree with the letter of what you said. My contention is with the spirit of the presentation by Gelfand et al which I find wrong (it's a personal opinion). I think if you loose countable additivity the measure belongs in the garbage can. Also, cylinder set measures are the wrong concept. We should be talking about Borel probability measures. Glimm and Jaffe, perhaps influenced by Hida or Gelfand et al, made a similar methodological error. | |
| Dec 30, 2017 at 22:48 | comment | added | Pedro Lauridsen Ribeiro | Gel'fand's approach actually considers more general (topological duals of nuclear) spaces too, but you may loose countable additivity for cylinder set measures. This is true for Schwartz's approach as well. However, you do get countable additivity just like in the case of nuclear countably Hilbert spaces if the (underlying TVS of the) measure space is the topological dual of a countable inductive limit of countably normed nuclear spaces - which includes $\mathscr{D}'$. In fact, Section IV.2 of Gel'fand-Vilenkin discusses all relevant cases for countable additivity of cylinder set measures. | |
| Dec 30, 2017 at 21:19 | comment | added | Abdelmalek Abdesselam | One issue with the Gelfand approach is that it only talks about countably Hilbert nuclear spaces. Grothendieck's definition is more general and probability theory should work well in this more general setting too. | |
| Dec 28, 2017 at 23:08 | history | answered | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |