Is there a simple way to construct such a measure? Preferably, one invariant under rotations and reflections of $R^N$.
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$\begingroup$ 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 $S^n$) $\endgroup$– user8268Mar 31, 2011 at 21:20
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$\begingroup$ 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? $\endgroup$– user8960Mar 31, 2011 at 22:05
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2 Answers
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To generate a uniformly distributed random point on $S^n$, one can generate a random vector $X=(X_k)_{1\leqslant k\leqslant n+1}$ with any nondegenerate isotropic distribution in $\mathbb R^{n+1}$, and compute the radius $R=\sqrt{X_1^2+X_2^2+\cdots+X_{n+1}^2}$. Then the vector $R^{-1}X$ is uniformly distributed on $S^n$.
The commonest choice of an isotropic distribution is the centered normal distribution with covariance matrix (any nonzero multiple of) the identity matrix. In other words, the coordinates of $X$ may be $n+1$ independent standard random variables.
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There is a unique probability measure on $P^n$ invariant under the "linear" action of $O(n+1,\mathbb R)\subset GL(n+1,\mathbb R)$. This follows from the existence of Haar measures on homogeneous spaces and is proved in this generality in pretty much any good textbook which constructs the Haar measure on groups —Lang's book on functional analysis is one.
answered Mar 27, 2012 at 20:36
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$\begingroup$ Books whose titles contain «integral geometry» are also good candidates. $\endgroup$ Mar 27, 2012 at 20:39