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Nonstandard integration theory in topological vector lattices. (English) Zbl 0890.46059

Summary: This paper develops a Daniell-Stone integration theory in topological vector lattices. Starting with an internal, vector valued, positive linear functional \(I\) on an internal lattice of vector valued functions, we produce a nonstandard hull valued integral \(J\) satisfying the monotone convergence theorem. Nonstandard hulls form a natural extension of infinite dimensional spaces and are equivalent to Banach space ultrapower constructions. The first application of our integral is a construction of Banach limits for bounded, vector valued sequences. The second example yields an integral representation for bounded and quasibounded harmonic functions similar to that of the Martin boundary. The third application uses our general integral to extend the Bochner integral.

MSC:

46S20 Nonstandard functional analysis
46G10 Vector-valued measures and integration
46A40 Ordered topological linear spaces, vector lattices
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28E05 Nonstandard measure theory
03H05 Nonstandard models in mathematics
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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References:

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