If f is a bijection and is a non-negative and bounded then the limit exists and is equal to , where the limit is over all integer intervals
Suppose that for all t. If I is an interval such that |I|>B then any infinite orbit intersects I. The sum of over the points in I lying in a given infinite orbit is bounded above by I and below by |I|-2B.
If I is large enough then the sum of for can be made arbitrarily close to the number of infinite orbits of f; the singleton orbits don't contribute since for those orbits. Thus in the limit the average of df over an interval of consecutive integers must become arbitrarily close to the number of infinite orbits of the permutation.