THE BEHAVIOR AT INFINITY OF AN INTEGRABLE FUNCTION

Abstract. Given a density d de\u85ned on the Borel subsets of [0;1); the limit at in\u85nity 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\u85es this type of be-havior at in\u85nity 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\u85nite series in regard to which the analogy between in\u85nite 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 f(x) = 1X n=1 [n;n+2n)(x); x 2 [0;1): Surprisingly, the analogy can be re-established if the usual limit is replaced by limit in density. This fact is implicit in a famous paper by B. O. Koopman and J. von Neumann [6] dedicated to weakly mixing transformations, and was recently made explicit and extended by us [10] to a scale of densities measuring how thin are the various Borel subsets of R. The aim of the present note is to provide a short argument for this general result along Koopman-von Neumanns ideas. In what follows we shall adopt the convention used in dynamical system theory for the iterates of a function f = f(x); f (0)(x) = x and f (n)(x) = (f f f | {z}) n times (x) for n 1: Note that f (n)(x) does not mean the nth derivative of f(x); a function that we never use in this paper