Timeline for Complete Boolean algebra not isomorphic to a $\sigma$-algebra
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 15, 2018 at 12:37 | history | edited | Martin Sleziak |
added (boolean-algebras) tag
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| Feb 25, 2014 at 8:54 | comment | added | alpha | It would be a pity not to mention the "Richtungsstreit" amongst measure theoreticians due to the distinction between the concepts of a $\sigma$ algebra or a complete Boolean algebra. The two approaches are measure spaces with points as the basic elements and measure algebras with events. We are using the terminology of David Fremlin who discusses the two approaches in his book "Topological Riesz Spaces" and his monumental five volume tome on measure theory (in particular, chapter 32) where he makes a lucid case for the second, less popular approach. | |
| Feb 22, 2014 at 5:49 | comment | added | Eric Wofsey | Every complete Boolean algebra is the regular open sets in some topological space, so Jochen's suggestion doesn't actually say much. | |
| Feb 22, 2014 at 1:27 | answer | added | François G. Dorais | timeline score: 12 | |
| Feb 21, 2014 at 21:02 | vote | accept | Bjørn Kjos-Hanssen | ||
| Feb 21, 2014 at 20:55 | answer | added | Joseph Van Name | timeline score: 15 | |
| Feb 21, 2014 at 19:49 | comment | added | Todd Trimble | Actually, your proposal is what occurred to me too, and I think with an extra nudge it can be made into an answer as well. | |
| Feb 21, 2014 at 19:32 | comment | added | Jochen Wengenroth | I did not claim to answer the question, I just can't imagine a Boolean isomorphism with some $\sigma$-algebra. | |
| Feb 21, 2014 at 19:24 | comment | added | Todd Trimble | @JochenWengenroth Is that actually enough? The question is whether there isn't a possibly exotic embedding of that Boolean algebra $B$ onto a $\sigma$-algebra supported on some other set $X$ (one preserving Boolean algebra structure and countable intersections), not necessarily the underlying set of the topological space you started with. | |
| Feb 21, 2014 at 19:21 | answer | added | Joel David Hamkins | timeline score: 7 | |
| Feb 21, 2014 at 19:17 | comment | added | Jochen Wengenroth | A candidate would be the complete Boolean algebra of regular open sets ($A$ is the interior of its closure) in $\mathbb R$ or some more exotic topological space. As open intervals are regular open the class itself is clearly not a $\sigma$-algebra. | |
| Feb 21, 2014 at 19:02 | answer | added | Simon Henry | timeline score: 21 | |
| Feb 21, 2014 at 17:51 | history | asked | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |