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Is there a simple way to construct such a measure? Preferably, one invariant under rotations and reflections of $R^N$.

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    If it is invariant wrt. reflections of $\mathbb{R}^n$ then its also invariant wrt. translations (as they are compositions of 2 reflection), so such a measure could not be finite. (If you wish just any reasonably natural measure, $\mathbb{RP}^n$ is $S^n/C_2$, so you can take the standard measure on $S^n$)2011-03-31
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    Thanks, user8960. Yes, if there is a standard measure on the sphere $S^N$ invariant under reflections through hyperplanes through the center and under rotations about the center, that's great. Is there a website or reference with the details of the construction of this measure?2011-03-31

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