Abstract
The classical Bochner integral is compared with the McShane concept of integration based on Riemann type integral sums. It turns out that the Bochner integrable functions form a proper subclass of the set of functions which are McShane integrable provided the Banach space to which the values of functions belong is infinite-dimensional. The Bochner integrable functions are characterized by using gauge techniques. The situation is different in the case of finite-dimensional valued vector functions.
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Schwabik, S., Guoju, Y. On the Strong McShane Integral of Functions with Values in a Banach Space. Czechoslovak Mathematical Journal 51, 819–828 (2001). https://doi.org/10.1023/A:1013721114330
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DOI: https://doi.org/10.1023/A:1013721114330