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
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 3, 2022 at 4:38 | history | edited | dohmatob | CC BY-SA 4.0 |
added 5 characters in body
|
| Apr 30, 2022 at 12:44 | comment | added | dohmatob | I guess you were referring to something like Theorem 3.11 of this paper citeseerx.ist.psu.edu/viewdoc/…. | |
| Apr 30, 2022 at 12:16 | comment | added | dohmatob | Thanks for the input. In fact in my original question $A$ was compact and convex (the unit probability simplex in $\mathbb R^k$). I diidn't know convexity would play in the problem, and so I suppressed it. Concerning the equality of the Aumann and Debreu integrals, is there are clear reference for this result ? (N.B.: I'm only learning all of this set-valued analysis here on the fly based on these interactions). Thanks. | |
| Apr 30, 2022 at 12:08 | comment | added | Martin Väth | (continuing comment): Note that the Debreu integral is defined by means of approximation of the mutlivalued functions in the hyperspace of nonempty compact (convex?) sets with the Hausdorff metric, hence by definition is compact (and convex). All result establishing this equality are one sense or another based on the weak compactness in $L_1$ of the set of selections of an integrably bounded multivalued map with compact convex values. I am afraid that for this result convexity is crucial. | |
| Apr 30, 2022 at 12:06 | comment | added | Martin Väth | As mentioned in a comment to my reply, I had a direct application of the measure isomorphism theorem in mind (which requires separability). However, I think that simply the proof of the convexity result by Aumann requires only a non-atomaic measure space, that is without loss of generality you can assume in a non-atomic measure space that $A$ is convex, and then again Aumann's proof for compactness should directly hold. Alternatively, once you assume that $A$ is convex, you can apply some of many results which establish equality of the Aumann and the Debreu integral. | |
| Apr 29, 2022 at 19:12 | history | edited | dohmatob | CC BY-SA 4.0 |
added 60 characters in body
|
| Apr 29, 2022 at 19:01 | history | edited | dohmatob | CC BY-SA 4.0 |
added 158 characters in body
|
| Apr 29, 2022 at 16:47 | history | edited | dohmatob | CC BY-SA 4.0 |
added 12 characters in body
|
| Apr 29, 2022 at 16:41 | history | edited | dohmatob | CC BY-SA 4.0 |
deleted 87 characters in body
|
| Apr 29, 2022 at 14:05 | history | edited | dohmatob | CC BY-SA 4.0 |
added 14 characters in body
|
| Apr 29, 2022 at 14:00 | history | edited | dohmatob | CC BY-SA 4.0 |
added 14 characters in body
|
| Apr 29, 2022 at 13:53 | history | edited | dohmatob | CC BY-SA 4.0 |
added 96 characters in body
|
| Apr 29, 2022 at 13:36 | history | edited | dohmatob | CC BY-SA 4.0 |
added 96 characters in body
|
| Apr 29, 2022 at 13:27 | history | answered | dohmatob | CC BY-SA 4.0 |