Timeline for Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel?
Current License: CC BY-SA 4.0
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| Dec 29, 2022 at 16:54 | comment | added | Akira | Thank you so much again. It seems the idea is to build a coarser topology that induces the same Borel $\sigma$-algebra. With coarser topology, we do not have enough open sets to approximate Borel sets. | |
| Dec 29, 2022 at 16:45 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Dec 29, 2022 at 16:44 | comment | added | Iosif Pinelis | @Akira : I have added such a modification. However, at this point I don't have a counterexample with a $T_1$ topology over $X$, so that every single-point subset of $X$ be a closed set. | |
| Dec 29, 2022 at 16:37 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Dec 29, 2022 at 16:35 | vote | accept | Akira | ||
| Dec 29, 2022 at 16:35 | comment | added | Akira | Thank you so much for your answer. I got it. Can we have an example where there is some $A \in \tau$ such that $\mu (A) \in (0, \infty)$? | |
| Dec 29, 2022 at 16:28 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |