Kantorovich–Wright integral and representation of quasi-Banach lattices

Abstract

The purpose of this paper is two-fold: first, to outline a purely order-based integral of the type of the Kantorovich–Wright integral of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a K σ-space (that is, a Dedekind σ-complete vector lattice) and, secondly, prove new theorems on the representation of Dedekind complete vector lattices and quasi-Banach lattices in the form of lattices of functions integrable or “weakly” integrable with respect to an appropriate vector measure. In particular, it is shown that, in studying quasi-Banach lattices, when the duality method does not apply, the Kantorovich–Wright integral is more flexible than the Bartle–Dunford–Schwartz integral.