The Behavior at Infinity of an Integrable Function

Given a density d de…ned on the Borel subsets of [0;1); the limit at in…nity in density of a function f : [0;1) ! R is zero if each of the sets ft : jf(t)j "g has zero density whenever " > 0: It is proved that every Lebesgue integrable function f : [0;1) ! R veri…es this type of behavior at in…nity with respect to a scale of densities including the usual one, d(A) = limr!1 m(A\[0;r)) r : The analogy between convergent series and integrals over the positive semi-axis was an elegant and fruitful subject present in all major treatises of mathematical analysis published during the 20th Century. As was noted by G. H. Hardy in his Course of Pure Mathematics [4], p. 324, there is one fundamental property of a convergent in…nite series in regard to which the analogy between in…nite series and in…nite integrals breaks down. If P an is convergent then an ! 0; but it is not always true, even when f : [0;1)! R is positive, that if R1 0 f(x)dx is convergent then f(x)! 0 as x!1. Due to the prominent role played by negligible sets one might expect that a conclusion of the type f(x)! 0 as x runs to 1 outside a negligible set must be working. That this is not the case is shown by the integrable function