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Zeitschrift für Analysis und ihre Anwendungen

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Volume 9, Issue 6, 1990, pp. 481–501
DOI: 10.4171/ZAA/419

Published online: 1990-12-31

Integration by Means of Riemann Sums in Banach Spaces I

Wolfgang Erben[1] and Gerhard Grimeisen

(1) Hochschule für Technik Stuttgart, Germany

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].

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Erben Wolfgang, Grimeisen Gerhard: Integration by Means of Riemann Sums in Banach Spaces I. Z. Anal. Anwend. 9 (1990), 481-501. doi: 10.4171/ZAA/419