generic points, the asymptotic average shadowing property, dbarpseudometric.

It is proved that to every invariant measure of a compact dynam- ical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for the measure. It follows that the asymptotic average shadowing property implies that every invariant measure has a generic point. The proof is based on the properties of the Besicovitch pseudometric DB which are of independent in- terest. It is proved among the other things that the set of generic points of ergodic measures is a closed set with respect to DB. It is also showed that the weak specification property implies the average asymptotic shadowing prop- erty thus the theory presented generalizes most known results on the existence of generic points for arbitrary invariant measures.