Timeline for Measures on a unit sphere of a Hilbert space
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 26, 2016 at 17:01 | comment | added | ijon | @NateEldredge Thank you. I will try to use projections of Gaussian measures. | |
| Jun 26, 2016 at 16:47 | comment | added | Nate Eldredge | There are certainly lots of measures other than your countably supported examples (for instance take a Gaussian measure and project to the sphere) but all of them bias some directions over others. In the "non-Borel" direction, I think it might be possible to do something with the algebra of "cylindrical sets" which are determined by finitely many coordinates. You can probably get a finitely additive measure on this algebra, but it won't be countably additive. | |
| Jun 26, 2016 at 16:22 | comment | added | ijon | It is hard to believe that the notion of a random direction in a Hilbert space have never been considered before. Since there is no "rotation" invariant measure on $S$ I will be grateful for other suggestions that can make the notion of "random direction" in a Hilbert space meaningful and natural. | |
| Jun 26, 2016 at 16:17 | comment | added | ijon | @NateEldredge I am dealing with a model of some system and the states of that system are represented as directions in a Hilbert space (or points on its unit sphere). I am looking for nice probabilistic measures on the unit sphere to make the notion of "random state" meaningful. I would also like to integrate some real functions on the set of all states of the system. Taking a measure concentrated on some countable subset of the set of all directions (states) is an option but it seems quite hard to justify any particular choice. That is why I am looking for some "canonical" ("natural") measure | |
| Jun 26, 2016 at 15:40 | comment | added | Nate Eldredge | I think my argument shows that no, there isn't. The space is too big. You can compare with the well-known fact that there isn't any Lebesgue measure on $H$ (any nontrivial translation-invariant Borel measure on $H$ gives infinite measure to every open set), or the fact that there isn't any Gaussian measure on $H$ whose covariance operator is the identity. I am not sure what you would be looking for in a "non-Borel" measure. | |
| Jun 26, 2016 at 12:13 | comment | added | ijon | @NateEldredge Ok so we can always pick countably many points on the sphere $S$ and assign weights to them. The problem with this approach is that such a choice is always arbitrary. Suppose I want to choose a random direction in a Hilbert space $H$, which is the same as picking a random point from $S$. Isn't there a "canonical", "natural" way of doing that? Isn't there really any "canonical" probabilistic (maybe even non-Borel) measure on a sphere $S$? | |
| Jun 25, 2016 at 14:49 | comment | added | Nate Eldredge | Well, there's counting measure. But it should be pretty easy to show that such $\mu$ must assign infinite measure to every open set (unless it is the zero measure). Given an open set $U$, find a small ball inside it which has positive measure, then rotate this ball a small amount about infinite many orthogonal axes so that the images stay inside $U$ but are disjoint. | |
| Jun 25, 2016 at 14:33 | review | First posts | |||
| Jun 25, 2016 at 14:50 | |||||
| Jun 25, 2016 at 14:30 | comment | added | Uri Bader | related: mathoverflow.net/q/234249/89334 | |
| Jun 25, 2016 at 14:24 | history | asked | ijon | CC BY-SA 3.0 |