  It is proved that to every invariant measure of a compact dynamical 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 interest. 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 property thus the theory
presented generalizes most known results on the existence of generic points for
arbitrary invariant measures.
