  We consider finite point subsets (distributions) in compact metric spaces. In
the case of general rectifiable metric spaces, non-trivial bounds for sums of
distances between points of distributions and for discrepancies of
distributions in metric balls are given (Theorem 1.1). We generalize
Stolarsky's invariance principle to distance-invariant spaces (Theorem 2.1).
For arbitrary metric spaces, we prove a probabilistic invariance principle
(Theorem 3.1). Furthermore, we construct equal-measure partitions of general
rectifiable compact metric spaces into parts of small average diameter (Theorem
4.1). This version of the paper will be published in Mathematika
