  We prove that for every ergodic invariant measure with positive entropy of a
continuous map on a compact metric space there is $\delta>0$ such that the
dynamical $\delta$-balls have measure zero. We use this property to prove, for
instance, that the stable classes have measure zero with respect to any ergodic
invariant measure with positive entropy. Moreover, continuous maps which either
have countably many stable classes or are Lyapunov stable on their recurrent
sets have zero topological entropy. We also apply our results to the Li-Yorke
chaos.
