Is there a simple way to construct such a measure? Preferably, one invariant under rotations and reflections of $R^N$.
A "natural" Borel probability measure on a projective space $P R^{N}$?
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0Thanks, 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
2 Answers
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.
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.
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0Books whose titles contain «integral geometry» are also good candidates. – 2012-03-27