Integration for positive measures with values in quasi-Banach lattices

The paper aims to overview some new ideas and recent results in the theory of integration of scalar functions with respect to a vector measure, as well as general theorems on the functional representation of quasi-Banach lattices. We outline a purely order-based Kantorovich–Wright type integral of scalar functions with respect to a vector measure defined on a $\delta$-ring and taking values in a Dedekind $\sigma$-complete vector lattice. The parallel Bartle–Dunford–Schwartz type integration with respect to a measure defined on a $\delta$-ring with values in a quasi-Banach lattice is also presented. In the context of Banach lattices a crucial role is played by the spaces of integrable and weakly integrable functions with respect to a vector measure. Dealing with the functional representation of quasi-Banach lattices a duality based approach does not work but there are two natural candidates for a space of weakly integrable functions: maximal quasi-Banach extension and the domain of the smallest extension of the integration operator. Using this idea, one can construct new spaces of weakly integrable functions that play an essential role in the problem of the functional representation of quasi-Banach lattices. In particular, it is shown that, in studying quasi-Banach lattices, when the duality method is inapplicable, the Kantorovich–Wright integral turns out to be more flexible than t more flexible than the Bartle–Dunford–Schwartz integral.