Timeline for Comparing two different principles of premeasure-to-measure extension
Current License: CC BY-SA 4.0
14 events
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| Nov 27 at 18:25 | comment | added | Atom | Okay, I will ask a new question if I hit a wall thinking about it. In any case, for all your time and responses, many thanks! :D | |
| Nov 27 at 18:08 | comment | added | Iosif Pinelis | @Atom : Sorry, I do not engage in such chat. You may want to ask further questions about this answer right here. Actually, the question about the completion does not concern this answer directly. However, on the second thought, the Wikipedia claim is correct, and it follows by mentioned Theorem 1.36 in Rudin's book. | |
| Nov 27 at 9:05 | comment | added | Atom | Let us continue this discussion in chat. | |
| Nov 27 at 8:52 | comment | added | Atom | Ah, my bad! I meant $\sigma(\mathscr A\cup\{ \text{subsets of $m$-null sets of $\sigma(\mathscr A)$}\} )$ (which is the set $\{A\cup N : A\in\sigma(\mathscr A)\text{ and $N$ is a subset of an $m$-null set in $\sigma(\mathscr A)$}\}$). | |
| Nov 27 at 3:27 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Nov 26 at 21:43 | comment | added | Iosif Pinelis | Previous comment continued: To see why the definition of the completion in my paper is equivalent to the one given in Theorem 1.36 in Rudin's book, you may want to use the last sentence in Proposition 2.3 in the paper. | |
| Nov 26 at 21:10 | comment | added | Iosif Pinelis | Previous comment continued: (iii) Your "standard completion" is not in general a $\sigma$-algebra. Perhaps, here you meant the claim "In the above construction [...]" in section Construction of a complete measure, but that claim is of course false. | |
| Nov 26 at 21:10 | comment | added | Iosif Pinelis | @Atom : Thank you for your appreciation of this answer. Some comments in response to yours: (i) The paper is published, with the content essentially identical to that of the preprint. (ii) The definition of the completion in the paper is standard. For instance, it is equivalent to the one given in Theorem 1.36 in Rudin's book. | |
| Nov 26 at 19:22 | comment | added | Atom | In the preprint, you also compare the "completion" $\mathscr M_{\mathsf{Co}}$ of $\sigma(\mathscr A)$, defined by $\{E\subseteq X\colon \exists S\in\sigma(\mathscr A)\ d_X(E,S)=0\}$. However, this "completion" is in general larger than the standard completion, $\sigma(\mathscr A)\cup\{\text{subsets of $m$-null sets}\}$, correct? | |
| Nov 26 at 19:04 | comment | added | Atom | What a complete answer! I am yet to read Proposition 1 above. I am going through your (very interesting!) preprint. | |
| Nov 26 at 18:59 | vote | accept | Atom | ||
| Nov 26 at 16:43 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Nov 26 at 14:54 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Nov 26 at 6:50 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |