A general rearrangement theorem for sequences (Q791918)

Let X be a compact metric space with metric d, and let \(\omega_ 1=(x_ n)\) and \(\omega_ 2=(y_ n)\) be sequences of elements of X. Then it is shown that the following two conditions are equivalent: (i) there exists a permutation \(\tau\) of the set of positive integers such that \(\lim_{n\to \infty}d(x_ n,y_{\tau(n)})=0\); (ii) \(\omega_ 1\) and \(\omega_ 2\) have the same accumulation points. From this result all the standard rearrangement theorems in the theory of uniform distribution of sequences can be obtained as simple consequences. For instance, it follows easily that if \(\mu\) is a Borel probability measure in X, then a sequence \(\omega\) of elements of X has a \(\mu\)-uniformly distributed (\(\mu\)-well distributed, completely \(\mu\)-uniformly distributed) rearrangement if and only if every point of the support of \(\mu\) is an accumulation point of \(\omega\).