Very often a measurement of a physical system takes the form of a finite
time average for some observable. For infinite time averages Birkhoff's
theorem classifies all the possible outcomes in terms of the invariant
measures of the system. The study of the, much more realistic, finite
time averages is equivalent to investigating at which speed the limit is
attained. This problem is only partially understood, essentially we
understand few special cases. Yet, our current knowledge shows that the
behaviour depends drastically from the properties of the system. In the
study of such a problem functional analysis, probability theory, and
geometry play major roles. I will attempt to give an overview of the
subject.