  Let X be a real or complex Hilbert space of finite but large dimension d, let
S(X) denote the unit sphere of X, and let u denote the normalized uniform
measure on S(X). For a finite subset B of S(X), we may test whether it is
approximately uniformly distributed over the sphere by choosing a partition
A_1,...,A_m of S(X) and checking whether the fraction of points in B that lie
in A_k is close to u(A_k) for each k=1,...,m. We show that if B is any
orthonormal basis of X and m is not too large, then, if we randomize the test
by applying a random rotation to the sets A_1,...,A_m, B will pass the random
test with probability close to 1. This statement is related to, but not
entailed by, the law of large numbers. An application of this fact in quantum
statistical mechanics is briefly described.
