Abstract integration with respect to measures and applications to modular convergence in vector lattice setting

TL;DR

This paper introduces a Bochner-type integral for vector lattice-valued functions with respect to vector lattice-valued measures, establishing fundamental properties and applying it to convergence of operators, moment kernels, and Brownian motion.

Contribution

It develops a new integral framework for vector lattice-valued functions and measures, enabling advanced convergence analysis and applications in stochastic processes.

Findings

Established properties of the vector lattice-valued integral.

Proved convergence results for operators in vector lattice-valued modulars.

Applied the integral to moment kernels and Brownian motion.

Abstract

A "Bochner-type" integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures is presented with respect to abstract convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion.