Timeline for answer to Non-separable metric probability space by Iosif Pinelis
Current License: CC BY-SA 4.0
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| Jul 31, 2018 at 16:05 | comment | added | Robert Furber | @IosifPinelis I see. The appendix must have been taken out of the second edition. This solves a minor mystery for me, because someone else I told about this a few years ago couldn't find that appendix either. | |
| Jul 31, 2018 at 0:41 | comment | added | Iosif Pinelis | @RobertFurber : Thank you for your very informative comments. They certainly deserve to be presented as a formal answer. The English original of the Billingsley book I saw at books.google.com/… , and I could not find Appendix III there. Apparently, what Appendix III mainly provides is Theorem 2, mentioned in my answer, stating that all probability measures on a metric space $X$ are separable iff every discrete subset of $X$ is of non-measurable cardinality. | |
| Jul 30, 2018 at 22:44 | comment | added | Robert Furber | My last comment -- Silver, Solovay and Kunen (see section 3.6 of Solovay's article Real-Valued Measurable Cardinals from the book Axiomatic Set Theory) proved that measurable cardinals, if consistent, are compatible with the generalized continuum hypothesis, and therefore the continuum hypothesis. So the answer to question 2 is no, because measurable cardinals are real-valued measurable. However, if the continuum hypothesis holds and there is a metric space $X$, with a non-separable probability measure on it, $X$ must have cardinality very much larger than the continuum. | |
| Jul 30, 2018 at 21:24 | comment | added | Robert Furber | One more thing -- the first measurable cardinal is weakly inaccessible, so the continuum is measure-free not only if $2^{\aleph_0} = \aleph_1$, but also if it is $\aleph_2$, $\aleph_3$, $\aleph_{\epsilon_0 + 1}$, $\aleph_{\omega_5}$, etc. | |
| Jul 30, 2018 at 21:14 | comment | added | Robert Furber | Billingsley also undersells the following point (as do all analysis books of that era when measurable cardinals come up, such as Gillman & Jerison's Rings of Continuous Functions and Schaefer's Topological Vector Spaces) -- it is not really an "unsolved problem" as to whether real-valued measurable cardinals exist, rather the existence of real-valued measurable cardinals proves the consistency of ZFC, so they cannot be proven to exist, nor even to be relatively consistent, if we just start with ZFC. A nice textbook reference for these facts is Jech's "Set Theory: Third Millennium Edition". | |
| Jul 30, 2018 at 21:10 | comment | added | Robert Furber | Additionally, the terminology Billingsley uses for measurable cardinals, although it is a direct translation of Ulam's terminology in the original paper, is out of date. Nowadays we say that a cardinal $\kappa$ is real-valued measurable if there is a $\kappa$-additive probability measure $\mu$ on $\kappa$, considered as a discrete metric space, such that $\mu(\{x\}) = 0$ for all $x \in \kappa$. It is then a theorem that the smallest cardinal with a countably additive probability measure vanishing at every point is the first real-valued measurable cardinal. | |
| Jul 30, 2018 at 21:00 | comment | added | Robert Furber | Appendix III, the reference to Keisler and Tarski, and Theorem 2 do appear in my copy of the 1968 original from Wiley and Sons. Which English original were you looking at? | |
| Jul 30, 2018 at 5:26 | comment | added | Aryeh Kontorovich | Many thanks for the detailed reply, Iosif. | |
| Jul 30, 2018 at 5:25 | vote | accept | Aryeh Kontorovich | ||
| Jul 29, 2018 at 23:02 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |