Timeline for Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mid \pi: X \to A\right\}$ closed?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 30, 2022 at 13:42 | comment | added | dohmatob | OK, thanks. I'ved added a post which captures my understanding of your solution (i.e providing low-level details) and your comments mathoverflow.net/a/421437/78539. | |
| Apr 29, 2022 at 21:01 | history | edited | Martin Väth | CC BY-SA 4.0 |
added 49 characters in body
|
| Apr 29, 2022 at 20:26 | comment | added | Martin Väth | By $v(\cdot)A$ I mean the multivalued function (which assumes values in the powerset of $\mathbb R^n$); the Aumann integral is defined as the set of integrals over all measurable selections of that multivalued function. Concerning going to Lebesgue measure: I had the theorem in mind that every atomless separable measure space is isomorphic to Lebesgue measure. I forgot to mention "separable". (One might still try to apply Maharam's results for the nonseparable case, but I am not sure whether this will lead to something.) | |
| Apr 29, 2022 at 13:59 | comment | added | dohmatob | I'ved added a post here mathoverflow.net/a/421367/78539 trying to expand on my understanding of the above post (Martin Vath's answer). | |
| Apr 29, 2022 at 11:19 | comment | added | dohmatob | @MichaelGreinecker Thanks for the insight. My issue is that i don't how to move from Lebesgue (the case treated in the reference paper) to general non-atomic probability measures. But I might be missing something. (BTW, I started a thread on this issue here mathoverflow.net/q/421356/78539) | |
| Apr 29, 2022 at 11:16 | comment | added | Michael Greinecker | @dohmatob You can decompose $X$ into a countable set on which $P$ is purely atomic and a set on which $P$ is nonatomic. You have a solution for both sets, the Minkowski sum will be again compact and give you your answer. | |
| Apr 29, 2022 at 9:47 | comment | added | dohmatob | Also, by $v(\cdot)A$ you meant $v(\cdot )\Pi$, right ? That is, the collection of functions of the form $x \mapsto v(x)\pi(x)$ with $\pi \in \Pi$. | |
| Apr 29, 2022 at 9:38 | comment | added | dohmatob | Thanks for the nice insight (upvoted). Indeed, it seems Theorem 4 would establish the result for Lebesgue measure. How does one go from non-atomic to general measures on $R^d$ ? Is it via some kind of representation theorem for non-atomic measures (on $R^d$) ? | |
| Apr 28, 2022 at 21:53 | history | edited | Martin Väth | CC BY-SA 4.0 |
added 1 character in body
|
| Apr 28, 2022 at 21:47 | history | edited | Martin Väth | CC BY-SA 4.0 |
deleted 56 characters in body
|
| Apr 28, 2022 at 21:38 | history | answered | Martin Väth | CC BY-SA 4.0 |