Timeline for A result of Sierpiński on non-atomic measures
Current License: CC BY-SA 3.0
17 events
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| Oct 11, 2023 at 6:14 | comment | added | Linas | Wikipedia says this: "The Cantor set is sometimes regarded as "universal" in the category of compact metric spaces, since any compact metric space is a continuous image of the Cantor set ... is sometimes known as the representation theorem for compact metric spaces." Using this, you now have the task of jumping between metric spaces and measures on sigma algebras. I think this step is discussed in books on General Topology. | |
| Oct 11, 2023 at 5:57 | comment | added | Linas | There is a theorem, (I don't recall the name; I think a woman stated it) also from early 20th century, that every "arbitrary set $S$" is the image of a product of Cantor sets, times a countable number of discrete points. There's an equivalent theorem for C* algebras. Intuitively (but imprecisely) it says that all sets look like subsets of $\mathbb{R}^n$, in the end. I think this became "common knowledge" in the first half of the 20th century, through a series of detailed proofs. But I don't know those proofs, and can't do the archeology to find them. | |
| Oct 5, 2023 at 18:10 | comment | added | Salvo Tringali | @Linas Sorry for the late reply, I've just seen your message. I don't quite remember what I had in mind back in 2015. But Thm 2 is about real-valued finitely additive measures defined on the whole power set of a bounded subset of $\mathbb R^n$. Thm 1 is about a non-atomic measure on an arbitrary set $S$. How do you obtain the latter from the former, considering that the kind of measures involved in the two statements are so different from each other? | |
| Sep 13, 2020 at 5:58 | comment | added | Linas | Doesn't one get thm 1 from thm 2 simply by picking some $E_2$ that has measure zero? It says "for all" and so surely I can pick something? Is this question secretly about the axiom of choice, or am I missing something that was supposed to be obvious? | |
| Dec 10, 2015 at 9:01 | comment | added | Salvo Tringali | Incidentally, the proof of the theorem (and, more specifically, its dependency on the axiom of choice) is discussed here: mathoverflow.net/questions/225677/…. | |
| Dec 7, 2015 at 22:40 | vote | accept | Salvo Tringali | ||
| Nov 10, 2015 at 21:12 | answer | added | Salvo Tringali | timeline score: 7 | |
| Nov 4, 2015 at 12:02 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Added an update and fixed a detail in the notes
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| Nov 4, 2015 at 11:11 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Added an update and fixed a detail in the notes
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| Nov 3, 2015 at 23:13 | history | edited | user9072 |
edited tags
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| Nov 3, 2015 at 22:58 | answer | added | Gerald Edgar | timeline score: 4 | |
| Nov 3, 2015 at 22:48 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Fixed a couple of mistakes in Note (*)
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| Nov 3, 2015 at 22:43 | history | edited | Gerald Edgar | CC BY-SA 3.0 |
misprint
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| Nov 3, 2015 at 22:22 | comment | added | Salvo Tringali | For the record: Theorem 1 is mentioned as well in a 2014 question by @MannyReyes: mathoverflow.net/questions/187975, which also makes reference to the same Wikipedia article as in the OP. | |
| Nov 3, 2015 at 22:08 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Added a couple of notes
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| Nov 3, 2015 at 22:03 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Added a couple of notes
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| Nov 3, 2015 at 21:41 | history | asked | Salvo Tringali | CC BY-SA 3.0 |