Consider a real separable infinite-dimensional Hilbert space $H$. Let $S=\{h\in H\mid \|h\|=1\}$ be a unit sphere in $H$. What are the most natural measures on $S$? Is there a (Borel) measure $\mu$ on $S$ which is invariant under the action of all orthogonal operators $H\rightarrow H$ on $S$? How about being invariant under the action of other interesting groups of operators?
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1$\begingroup$ related: mathoverflow.net/q/234249/89334 $\endgroup$– Uri BaderCommented Jun 25, 2016 at 14:30
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2$\begingroup$ 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. $\endgroup$– Nate EldredgeCommented Jun 25, 2016 at 14:49
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$\begingroup$ @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 $S$? $\endgroup$– ijonCommented Jun 26, 2016 at 12:13
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$\begingroup$ 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. $\endgroup$– Nate EldredgeCommented Jun 26, 2016 at 15:40
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$\begingroup$ @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 $\endgroup$– ijonCommented Jun 26, 2016 at 16:17
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