Timeline for Relation between projective hierarchy and universally measurable sets
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 18, 2015 at 14:22 | comment | added | yada | Oh yes, of course: replace "countably many" by "continuum many" (which doesn't change your computations). It is really interesting to see the huge gap between the definable sets (projective hierarchy, defined from "inner" by unions) and the universally measurable sets (defined from "outer" by cutting, i.e. intersections of $\sigma$-algebras). | |
| Dec 18, 2015 at 14:10 | comment | added | Joel David Hamkins | I think of the projective sets in terms of an $\omega$ hierarchy, taking projections and complements, rather than an $\omega_1$-hierarchy. Alternatively, one may think of the defining formulas, of which there are only countably many. If one allows real parameters, this gives continuum many boldface projective sets. | |
| Dec 18, 2015 at 14:07 | vote | accept | yada | ||
| Dec 18, 2015 at 14:07 | comment | added | yada | Thank you. I haven't thought about the fact that the projective hierarchy defines at most countably many new sets in each stage of the construction (similarly to the Borel hierarchy). Since the induction runs up to $\omega_1$ stages, there are only countably many projective sets. | |
| Dec 18, 2015 at 13:48 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |