Timeline for Quotients of Measurable Spaces?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 19, 2022 at 18:32 | comment | added | Michael Greinecker | @AidanYoung "On the fundamental ideas of measure theory". It's more of a booklet though; 54 pages in total. | |
| May 19, 2022 at 18:09 | comment | added | Aidan Young | @MichaelGreinecker The link for the Rohlin book isn't working. What was the book? | |
| Jul 18, 2014 at 11:48 | comment | added | SBF | I think this question of mine is related, though I am not sure whether the maximal/minimal $\sigma$-algebra would work there. | |
| Feb 8, 2012 at 20:03 | comment | added | Clinton Conley | You are right; I meant to say "restriction of a standard Borel $\sigma$-algebra." I'm not having good luck posting coherently lately. | |
| Feb 8, 2012 at 19:59 | comment | added | Clinton Conley | A good default reference for properties of standard Borel spaces would be Kechris' descriptive set theory text. More recent references about equivalence relations in particular include Kanovei's Borel Equivalence Relations and Gao's Invariant Descriptive Set Theory. Also noteworthy is the Jackson, Kechris, Louveau paper entitled "Countable Borel equivalence relations," J. Math. Logic. | |
| Feb 8, 2012 at 19:56 | comment | added | Michael Greinecker | A countable generated $\sigma$-algebra that separates points is not necessarily standard Borel. If the continuum hypothesis is wrong, we can take the trace $\sigma$-algebra of the Borel $\sigma$-algebra of an uncountable set of real numbers with cardinality less than $\mathfrak{c}$. | |
| Feb 8, 2012 at 19:45 | comment | added | Michael Greinecker | Thank you! Could you recommend a reference (as a starting point) for these kinds of results? | |
| Feb 8, 2012 at 19:44 | comment | added | Clinton Conley | Note that if your original $\Sigma$ is countably generated and separates points, it is automatically standard Borel, so the previous comment applies. | |
| Feb 8, 2012 at 19:36 | comment | added | Clinton Conley | In the special case that $\Sigma$ is standard Borel (generated by a Polish topology), this sort of question has been extensively studied among descriptive set theorists. For example, if $\Pi$ is a partition arising from a Borel equivalence relation, the quotient $\sigma$-algebra is countably generated if and only if there is a Borel assignment of real invariants to the equivalence classes. In the special case that the equivalence relation has countable classes (among other such special cases), this is equivalent to finding a Borel set which intersects each class in exactly one point. Etc. | |
| Feb 8, 2012 at 19:12 | comment | added | Michael Greinecker | Rohlin's book is very intersting. It can be found at: ma.huji.ac.il/~matang02/rohlin.pdf | |
| Feb 8, 2012 at 19:06 | history | edited | Michael Greinecker | CC BY-SA 3.0 |
added 220 characters in body
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| Jul 27, 2011 at 17:45 | comment | added | Gerald Edgar | Sometimes they just look at one of these as a counterexample. The quotient sigma-algebra for $\mathbb R / \mathbb Q$ . Or the "tail" sigma-algebra in a product $\prod_{n=1}^\infty T_n$ where the factors $T_n$ are nice. Or the "countable subsets of $\mathbb R$ ", realized as the sequences $\mathbb R^{\mathbb N}$ modulo the permutations. A point is: if it is not countably separated, then such a sigma-algebra is very bad in a sense that logicians will tell you about. | |
| Jul 27, 2011 at 17:18 | comment | added | Anthony Quas | Rokhlin's book "On the fundamental ideas of measure theory" develops a theory of measurable partitions... | |
| Jul 27, 2011 at 14:32 | history | asked | Michael Greinecker | CC BY-SA 3.0 |