Locally integrable functions and their indefinite integrals (Q2024017)

Let \(\mu\) be a locally determined positive measure on a measurable space \((\Omega,\Sigma)\) and consider the \(\delta\)-ring \(\Sigma^f=\{B\in \Sigma:\mu(B)<\infty\}\). Let \(X\) be a Banach space. A function \(F:\Omega \to X\) is called locally Bochner (resp., locally Pettis) integrable if \(\chi_B F\) is Bochner (resp., Pettis) integrable for every \(B\in\Sigma^f\). In this case, the map \(\Sigma^f \ni B \mapsto \int_B F \, d\mu \in X\) is a countably additive measure on \(\Sigma_f\) (called the indefinite integral of \(F\)). The paper under review contains some basic results on locally Bochner/Pettis integrable functions, their indefinite integrals, and the spaces of scalar functions which are integrable with respect to them.