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An extension of the Bochner integral generalizing the Loeb-Osswald integral. (English) Zbl 0891.28013

The author presents an interesting new approach to the definition of the extension of the Bochner integral. The underlying measure spaces are an internal finite measure space and its corresponding Loeb space. The functions under consideration are valued in a Banach space or in its non-standard hull. Integrating such functions is now defined by means of a three-step process: select an \(S\)-integrable internal lifting of the function, integrate the lifting internally and then project it on the nonstandard hull. In doing so, the author establishes a number of results from which we quote the following. If the Banach space is a Banach lattice, then the integral is the so-called Loeb-Osswald integral. A dominated convergence theorem is established. By constructing in an explicit manner a measure without a Radon-Nikodým derivative it is shown in a different manner that super-Radon-Nikodým implies superreflexivity.

MSC:

28E05 Nonstandard measure theory
46S20 Nonstandard functional analysis
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