# $L^1$-spaces of vector measures with vector density

## TL;DR

This paper explores the structure of $L^1$-spaces associated with vector measures that have vector densities, establishing connections with various integrability concepts like Pettis and Bochner integrals.

## Contribution

It introduces a framework linking $L^1$-spaces of vector measures with different integrability types, clarifying their relationships and properties.

## Key findings

- Identifies connections between $L^1_w(
u_F)$, $L^1(
u_F)$, and $L^1(|
u_F|)$ spaces.
- Provides conditions under which vector measures have densities with respect to positive measures.
- Clarifies the relationship between vector measure integrability and classical function spaces.

## Abstract

Let $F$ be a function with values in a Banach space. When $F$ is locally (Pettis or Bochner) integrable with respect to a locally determined positive measure, a vector measure $\nu_F$ with density $F$ defined on a $\delta$-ring is obtained. We present the existing connection between the spaces $L^1_w(\nu_F)$, $L^1(\nu_F)$ and $L^1(|\nu_F|)$ and the spaces of Dunford, Pettis or Bochner integrable functions.