Integration by Means of Riemann Sums in Banach Spaces I

Abstract

In 1938, G. Birkhoff [2] developed a theory of integration in Banach spaces which uses the approach via modified Riemann sums. In our paper, we generalize Birkhoff’s integration by basing it on an arbitrary set of "$\mu$-partitions" (being directed in a natural way) instead of the set of all "$\mu$-partitions" of the underlying measure space $(F,\mu )$. Our technique of working with infinite Riemann sums and limits of filtered families of such sums takes advantage of the 1-point completions of certain partial universal algebras as discussed by G. Grimeisen in [13].