Timeline for Measure theory on abstract Boolean ring
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 8, 2020 at 15:05 | vote | accept | Vasily Ilin | ||
| Jul 2, 2020 at 6:53 | comment | added | Dave L Renfro | See John M. H. Olmsted, Lebesgue theory on a Boolean algebra, Transactions of the American Mathematical Society 51 #1 (January 1942), pp.164-193 AND Roman Sikorski, The integral in a Boolean algebra, Colloquium Mathematicum 2 #1 (1949), pp. 20-26. | |
| Jul 2, 2020 at 1:08 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
deleted 4 characters in body; edited tags
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| Jul 2, 2020 at 1:05 | answer | added | Dmitri Pavlov | timeline score: 4 | |
| Jul 2, 2020 at 0:16 | comment | added | kirk sturtz | You want to look at Fred Lintons thesis and his paper "Functorial measure theory" in the Irvine Proceedings, Thompson , Washington D.C., !966. | |
| Jul 1, 2020 at 23:16 | comment | added | Vasily Ilin | @LSpice, yes, I did not think about what I am trying to integrate. Indeed, we need a function not on sets but on actual elements. YCor, I got confused in definitions. A sigma-algebra is a Z_2-algebra, which is also a Boolean ring, i.e. 2x = 0 and x^2 = x for every element x. Also, the point about countable additivity is good. | |
| Jul 1, 2020 at 23:03 | comment | added | Michael Greinecker | @LSpice You can identify a measurable function with the function that maps Borel sets to their preimages. This gives you a ring homomorphism from the Borel sets to the ring of measurable sets and you can build a theory of integration for such homomorphisms. | |
| Jul 1, 2020 at 22:55 | comment | added | YCor | You wrote "$\mathbf{Z}_2$-algebra (aka Boolean ring)" but obviously not every $\mathbf{Z}_2$-algebra is a Boolean ring. | |
| Jul 1, 2020 at 22:48 | comment | added | Michael Greinecker | Do you want your measure theory to be $\sigma$-additive? | |
| Jul 1, 2020 at 22:41 | comment | added | LSpice | What would you be trying to integrate? One integrates functions (which are not elements of the $\sigma$-algebra), not subsets. You could maybe build some formal theory (consider the completion of the group algebra of your ring in some appropriate topology …), but the Stone representation theorem says that any algebra is an algebra of subsets anyway, so it seems like you'd be quite close to doing measure theory while carefully avoiding naming the set. | |
| Jul 1, 2020 at 22:41 | history | edited | LSpice | CC BY-SA 4.0 |
\mathbb{Z_2} -> \mathbb Z_2 (2 shouldn't be bold)
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| Jul 1, 2020 at 22:19 | review | First posts | |||
| Jul 1, 2020 at 22:34 | |||||
| Jul 1, 2020 at 22:18 | history | asked | Vasily Ilin | CC BY-SA 4.0 |