On the set of limits of Riemann sums

Abstract

Let F map [0, 1] into a Banach space B and let R(F) denote the set of all limits of Riemann sums of F. The set R(F) need not be convex in general (Nakamura and Amemiya (1966)) but is always convex when B is finite dimensional as first shown by Hartman (1947). A proof of Hartman's result, based on a description of R(F) when the range of F is finite, was given in Ellis (1959). In this note this description is refined, the extreme points of R(F) are determined and the following complete characterization of R(F) is obtained (where Nn = {1,2, …, n}).