Timeline for Approximation by simple functions on a product $\sigma$-algebra
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jan 20, 2018 at 14:00 | comment | added | Pietro Majer | I think in general you need to iterate the point-wise limit uncountably many times to reach the sigma algebra generated. But if you have a measure, and you consider a.e. equivalence and a.e. convergence, then the monotone class theorem ensures that few iterations of "point-wise monotone limits" suffice. | |
| Jan 20, 2018 at 11:02 | answer | added | Michael Greinecker | timeline score: 3 | |
| Jan 16, 2018 at 14:21 | comment | added | 0xbadf00d | @PhoemueX Yes, this generalizes the functional monotone class theorem. I've tried to use it, but the problem is to show the closedness under bounded convergence. | |
| Jan 16, 2018 at 7:55 | comment | added | PhoemueX | I am not sure that what you want to show is true. What is true is that if you consider the smallest class containing $\mathcal{E}(...)$ and which is closed under bounded pointwise convergence, then it contains all bounded measurable (w.r.t. the product sigma algebra) functions. This follows from Dynkins multiplicative system theorem, see e.g. coursehero.com/file/p3097l/… | |
| Jan 16, 2018 at 2:35 | history | asked | 0xbadf00d | CC BY-SA 3.0 |