An extension of the Bochner integral generalizing the Loeb–Osswald integral

Abstract

In [10] Loeb and Osswald develop a Daniell–Stone integration theory in topological vector lattices. As an application they obtain an extension of the Bochner integral for functions with values in a topological vector lattice. The Daniell–Stone approach requires an order structure for definition of a null function. Here we build on the ideas in [10], but abandon the lattice structure. The internal representatives of the extended integrable functions in [10] turn out to be S-integrable liftings. This is our starting point: we define the class of extended integrable functions as the class of functions that have S-integrable liftings; a definition that is independent of any order structure. As a consequence, several of the proofs in the construction of the Banach space of extended integrable functions are almost identical to the analogous proofs in [10] and we omit them. After defining the integral we show that each extended integrable function induces a countably additive vector measure of bounded variation. This permits us to use the tools of the theory of vector measures. If the nonstandard hull of the Banach space has the Radon–Nikodym property, then the vector measure induced by an extended integrable function has a Bochner integrable Radon–Nikodym derivative; this yields a characterization of the Banach space of extended integrable functions. On the other hand for some spaces we can construct internal functions and standardize them to get very ‘spiky’ extended integrable functions whose induced measure is not differentiable on any set of positive measure. With this method we show that if a Banach space is not superreflexive, then it does not have the super-Radon–Nikodym property.