The behavior at infinity of an integrable function

Given a density dd defined on the Borel subsets of [0,∞)[0,∞), the limit at infinity in density of a function f:[0,∞)→Rf:[0,∞)→R is zero if each of the sets {t:|f(t)|≥ε}{t:|f(t)|≥ε} has zero density whenever ε>0ε>0. It is proved that every Lebesgue integrable function f:[0,∞)→Rf:[0,∞)→R verifies this type of behavior at infinity with respect to a scale of densities including the usual one, d(A)=limr→∞m(A∩[0,r))r.