Timeline for A question about the range of a positive measure
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 19, 2021 at 15:56 | answer | added | jacaboul | timeline score: 0 | |
| Oct 12, 2021 at 12:10 | history | edited | jacaboul | CC BY-SA 4.0 |
deleted 55 characters in body
|
| Oct 11, 2021 at 16:49 | vote | accept | jacaboul | ||
| Oct 10, 2021 at 8:15 | comment | added | jacaboul | Remember that I am considering the case where $\mu$ is atomless (or at least is not purely atomic). In that case once we know that $\sup\{\mu(A):A\in\cal C\}=r$, the proof that there is a set $A\in \cal M$ such that $\mu(A)=r$ is straightforward (see below). | |
| Oct 8, 2021 at 21:48 | answer | added | juan | timeline score: 3 | |
| Oct 8, 2021 at 19:49 | comment | added | Willie Wong | But your "simple path" can't possibly work. Consider the set $\{a, b, c, d\}$ with the counting measure. Start with $r = 3/2$. Your set $\mathcal{C} = \{\{a\}, \{b\}, \{c\}, \{d\}\}$ and is not closed under finite union. | |
| Oct 8, 2021 at 19:41 | comment | added | Willie Wong | What you sketched doesn't seem to prove what you want: if the range of $\mu$ were a dense proper subset of $[0,\mu(X)]$, it would satisfy $\sup \{ \mu(A): A\in \mathcal{C}\} = r$. | |
| Oct 8, 2021 at 18:17 | history | edited | jacaboul | CC BY-SA 4.0 |
added 1 character in body
|
| Oct 8, 2021 at 17:41 | comment | added | jacaboul | Also, it is true that the general case can be reduced to a large extent to the case of Lebesgue measure on an interval, but I would like to know if a more direct argument not using transfinite induction works in the abstract case. | |
| Oct 8, 2021 at 17:40 | comment | added | jacaboul | Thank you Christian for your answers. However, neither the wiki article nor the previous post on this site fullfill what I'm looking for. The proof proposed by wiki uses transfinite induction (Haussdorf's maximality theorem). For the other post on this site, it refers to an older proof due to Sierpinski that is constructive (and quite elegant, one of the first proofs of this kind I guess) but rather lengthy and works in R^n (allowing one to cut the measure space conveniently using regular grids). | |
| Oct 8, 2021 at 16:51 | review | Close votes | |||
| Oct 8, 2021 at 17:57 | |||||
| Oct 8, 2021 at 16:33 | comment | added | Christian Remling | Maybe this is helpful: en.wikipedia.org/wiki/… | |
| Oct 8, 2021 at 13:32 | history | asked | jacaboul | CC BY-SA 4.0 |