Timeline for The space of Borel function modulo comeager sets is Dedekind complete
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2019 at 8:26 | vote | accept | Littlefield | ||
| Mar 11, 2019 at 0:53 | answer | added | Robert Furber | timeline score: 3 | |
| Mar 8, 2019 at 14:11 | history | edited | YCor |
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| Mar 8, 2019 at 13:51 | answer | added | Gro-Tsen | timeline score: 1 | |
| Mar 8, 2019 at 12:43 | comment | added | Littlefield | @RobertFurber I would be happy if you give a reference for the proof or outline it in an answer. | |
| Mar 8, 2019 at 8:36 | comment | added | Robert Furber | @Littlefield You are right that Borel sets modulo meagre sets is always a complete Boolean algebra - I just couldn't see the right way of proving this at first, which is to enlarge the Borel $\sigma$-algebra to the $\sigma$-algebra of sets having the Baire property. | |
| Mar 8, 2019 at 8:35 | comment | added | Robert Furber | @Littlefield The set of measurable functions to $\mathbb{R}$ is not literally a complete lattice - for instance the sequence of constant functions $n : X \rightarrow \mathbb{R}$ has no least upper bound, because it has no upper bound. The actual statement is that every bounded set of functions has a least upper bound. By translating and rescaling, this is equivalent to $[0,1]$-valued functions being a complete lattice, and this is a little easier to prove technically. Unfortunately I do not know a reference, but I can outline the proof in an answer. | |
| Mar 8, 2019 at 8:21 | comment | added | Littlefield | @RobertFurber This result relating completeness of $\sigma$-ideals and quotient of measurable functions is really interesting and beautiful; thanks. Could you give me a reference of that result? I am pretty sure that the set of Borel sets modulo meager sets is a complete Boolean algebra, so this would solve my question. A question: why do you use [0,1] instead of $\mathbb R$? I suppose that the assertion is the same in both cases. | |
| Mar 7, 2019 at 23:04 | comment | added | Robert Furber | @Gro-Tsen I'll do that, but a bit later, because it's night time for me now. | |
| Mar 7, 2019 at 22:29 | comment | added | Gro-Tsen | @RobertFurber Even with whatever assumption is necessary to identify Borel-sets-modulo-meager-sets with regular open sets (even if $X$ is $\mathbb{R}^m$, say), it's still not clear to me why the space of Borel-functions-modulo-meager-sets that OP asks about can be identified (can it?) with the Dilworth space I mention in my first comment. (Can it?) I think you should post an answer, even if you need to make a few assumptions on $X$. | |
| Mar 7, 2019 at 21:01 | comment | added | Robert Furber | I should mention the following fact: If $(X,\Sigma)$ is a measure space equipped with a $\sigma$-ideal $\mathcal{N} \subseteq \Sigma$, then the poset of measurable functions $f : X \rightarrow [0,1]$ measurable with respect to the Borel $\sigma$-algebra on [0,1], identifying functions that differ only on a set from $\mathcal{N}$, is a complete lattice iff $\Sigma/\mathcal{N}$ is a complete Boolean algebra (instead of just $\sigma$-complete). With this result, the question is equivalent to the question of whether the Borel sets modulo the meagre sets is always a complete Boolean algebra.. | |
| Mar 7, 2019 at 20:41 | comment | added | Robert Furber | In the general, non-Baire-space case, I'm not certain that the space of Borel functions modulo meagre sets is Dedekind complete. A counterexample, if there is one, would have to be a space that is not Baire and not c.c.c.. | |
| Mar 7, 2019 at 20:37 | comment | added | Robert Furber | @Gro-Tsen You will need the space to be a Baire space for them to be the same. For instance, if $X$ itself is meagre (such as if $X = \mathbb{Q}$) then there is only one equivalence class of Borel functions modulo meagre sets, but there are non-trivial regular open sets (e.g. the strictly positive rationals). | |
| Mar 7, 2019 at 20:36 | comment | added | მამუკა ჯიბლაძე | @Gro-Tsen An answer on math.SE seems to be relevant | |
| Mar 7, 2019 at 18:56 | history | edited | Gro-Tsen |
add "real-analysis" tag
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| Mar 7, 2019 at 18:43 | comment | added | Gro-Tsen | (I suspect there is some relation, because the Boolean algebra of regular open sets, again under certain assumptions I'm not too sure about, coincides with the Boolean algebra of Borel sets modulo meager sets, cf. Halmos, "Lectures on Boolean Algebras" (1963), §13. But I'm afraid I'm unable to fill the dots.) | |
| Mar 7, 2019 at 18:38 | comment | added | Gro-Tsen | I have a followup question: Dilworth proves in his 1950 paper "The Normal Completion of the Lattice of Continuous Functions" that (possibly with assumptions on $X$ like T3½) the Dedekind-completion of real-valued continuous functions can be identified with (a)normal upper- (or lower-) semicontinuous real-valued functions on $X$, and (b)continuous real-valued functions on the Stone space of the (complete) Boolean algebra of regular open sets on $X$. What is the relation with the space in the question? | |
| Mar 7, 2019 at 18:21 | history | asked | Littlefield | CC BY-SA 4.0 |