Let X be an infinite-dimensional Banach space. It is well-known that the space of X-valued, Pettis integrable functions is not always complete with respect to the topology of convergence in mean, that is, the uniform convergence of indefinite integrals (see ). The Archimedes integral introduced in  does not suffer from this defect. For the Archimedes integral, functions to be integrated are allowed to take values in a locally convex space Y larger than the space X while X accommodates the values of indefinite integrals. Moreover, there exists a locally convex space Y, into which X is continuously embedded, such that the space ℒ(μX, Y) of Y-valued, Archimedes integrable functions is identical to the completion of the space of X valued, simple functions with repect to the toplogy of convergence in mean, for each non-negative measure μ (see ).