On the behavior of Lebesgue integrable functions at infinity

We investigate the behavior of sequences (f(cnx)) for Lebesgue integrable functions f:Rd→R. In particular, we give a description of classes of multipliers (cn) and (dn) such that f(cnx)→0 or ∑n=1∞|f(dnx)|<∞ for λ almost every x∈Rd.