An arithmetic property of Riemann sums

著者: Walter Rudin - Proceedings of the American Mathematical Society 1964 被引用: 27

holds for all real x if / is Riemann integrable on [0, l]. In the present note it is shown that there are bounded measurable functions / for which (2) is false for every x and that this convergence problem has some interesting number-theoretic aspects. In 1934, Jessen [l] proved that if/EL1 on [0, 1] and if {«*} is an increasing sequence of positive integers in which each term divides the next, then