THE ASYMPTOTIC BEHAVIOR OF INTEGRABLE FUNCTIONS

Given a density d defined on the Borel subsets of [0,infinity), the limit in density of a function f : [0, infinity) -> R is zero (abbreviated, (d)-lim(x ->infinity) f(x) = 0) if there exists a set S of zero density such that f(x) -> 0 as x runs to infinity outside S. It is proved that the behavior at infinity of every Lebesgue integrable function f is an element of L-1(0,infinity) satisfies the relations (d((n)))-lim(x ->infinity) (pi(n )(k=0)ln((k) )x) f(x) = 0, where (d((n)))(n) is a scale of densities including the usual one, d((0))(A) = lim(r ->infinity) m(A boolean AND[0,r))/r.