  The classical Birkhoff ergodic theorem in its most popular version says that
the time average along a single typical trajectory of a dynamical system is
equal to the space average with respect to the ergodic invariant distribution.
This result is one of the cornerstones of the entire ergodic theory and its
numerous applications. Two questions related to this subject will be addressed:
how large is the set of typical trajectories, in particular in the case when
there are no invariant distributions, and how the answer is connected to
properties of the so called natural measures (limits of images of "good"
measures under the action of the system).
