Timeline for Explicit examples of (probability) measures on $\prod \mathbb{R}$
Current License: CC BY-SA 4.0
12 events
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| Feb 11, 2020 at 13:54 | comment | added | Nate Eldredge | @MrMMS: Usually a good resource is cantorsattic.info, but it seems to be down at present. Otherwise there's Wikipedia. But as mentioned I am really not a good person to ask about such things. | |
| Feb 11, 2020 at 13:51 | comment | added | Nate Eldredge | @YCor: I really shouldn't have said that because I don't actually know what I'm talking about. All I mean is that in general I don't know of "interesting" ways to produce measures on such large spaces, certainly not any that give us such a rich class of measures as for countable products, and that the only examples I know of "large" spaces with really nontrivial measures are things like measurable cardinals. | |
| Feb 11, 2020 at 12:22 | comment | added | Robert Furber | @YCor I believe Nate Eldredge is alluding to the product measure extension axiom, or PMEA. This allows us to extend the measures described by Iosif Pinelis not only to the Borel $\sigma$-algebra of $\mathbb{R}^{\mathbb{R}}$, but all the way to the power set $\sigma$-algebra. PMEA implies there exists a real-valued measurable cardinal $\kappa < \mathfrak{c}$, and the consistency of PMEA is implied by the existence of a supercompact cardinal. | |
| Feb 11, 2020 at 9:44 | comment | added | YCor | @NateEldredge How can measurable cardinals help produce measures on $\prod_\mathbf{R}\mathbf{R}$ (which is much smaller than any measurable cardinal)? On the other hand this space is separable so it's easy to produce fully supported Borel probabilty measures on this space (e.g. atomic, but non-atomic too, by tensoring with a fully supported proba on $\mathbf{R}$. | |
| Feb 11, 2020 at 8:44 | comment | added | ABIM | @NateEldredge What is a measurable cadinal? (ie: a nice reference to them where I can read up?) | |
| Feb 11, 2020 at 8:29 | vote | accept | ABIM | ||
| Feb 10, 2020 at 22:53 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Feb 10, 2020 at 22:49 | comment | added | Iosif Pinelis | @NateEldredge : You are right. I have now corrected this. | |
| Feb 10, 2020 at 22:47 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 10 characters in body
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| Feb 10, 2020 at 22:33 | comment | added | Nate Eldredge | But AFAIK, if $T$ is uncountable then Kolmogorov does not give an extension to the Borel $\sigma$-algebra, only to the product $\sigma$-algebra, which is much smaller. I do not know of any way to produce nontrivial measures on the Borel $\sigma$-algebra of an uncountable product of $\mathbb{R}$, except using things like measurable cardinals. | |
| Feb 10, 2020 at 22:29 | comment | added | Nate Eldredge | Indeed, the Kolmogorov extension theorem is the main tool for producing probability measures on this space, and is probably the first thing the OP ought to study if not already familiar with it. | |
| Feb 10, 2020 at 22:25 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |