Kuelbs-Steadman spaces for Banach space-valued measures

TL;DR

This paper introduces Kuelbs-Steadman spaces for Banach space-valued measures, exploring their properties and embeddings in $L^p$-type spaces with respect to different modes of convergence.

Contribution

It extends Kuelbs-Steadman spaces to Banach space-valued measures and analyzes their main properties and embeddings, a novel generalization in the field.

Findings

Kuelbs-Steadman spaces can be constructed for Banach space-valued measures.

These spaces embed into classical $L^p$-spaces under certain conditions.

The paper characterizes the differences between norm and weak convergence in these spaces.

Abstract

We introduce Kuelbs-Steadman-type spaces for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate their main properties and embeddings in $L^p$-type spaces, considering both the norm associated to norm convergence of the involved integrals and that related to weak convergence of the integrals.