  We are now witnessing a rapid growth of a new part of group theory which has
become known as "statistical group theory". A typical result in this area would
say something like ``a random element (or a tuple of elements) of a group G has
a property P with probability p". The validity of a statement like that does,
of course, heavily depend on how one defines probability on groups, or,
equivalently, how one measures sets in a group (in particular, in a free
group). We hope that new approaches to defining probabilities on groups
outlined in this paper create, among other things, an appropriate framework for
the study of the "average case" complexity of algorithms on groups.
