Nonstandard integration theory in topological vector lattices

This paper develops a Daniell-Stone integration theory in topological vector lattices. Starting with an internal, vector valued, positive linear functionalI on an internal lattice of vector valued functions, we produce a nonstandard hull valued integralJ 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.