Isomorphismes entre des espaces de mesures \`a valeurs vectorielles

TL;DR

This paper establishes conditions under which various vector-valued measure spaces are isomorphic, focusing on the equivalence of these spaces when their underlying measure spaces are strongly isomorphic in the $L^1$ sense.

Contribution

It demonstrates that isomorphisms between measure spaces induce isomorphisms between associated vector measure and function spaces, extending the understanding of their structural relationships.

Findings

Isomorphisms between $L^1$ spaces imply isomorphisms of vector measure spaces.

The spaces $L^p( ext{measure}_1, X)$, $VB^p( ext{measure}_1, X)$, and $cabc( ext{measure}_1, X)$ are isomorphic to their counterparts when the underlying measures are strongly isomorphic.

The results unify the structure of vector measure spaces across different measure spaces under strong isomorphism conditions.

Abstract

Let $(\Omega_1, \mathcal{F}_1, \mu_1)$, $(\Omega_2, \mathcal{F}_2, \mu_2)$ be two probabilty spaces, $1\leq p\leq +\infty$ and $X$ a Banach space. In this work we show that $L^p(\mu_1, X)$, $VB^p (\mu_1,X),$ $cabv(\mu_{1},X)$ are isomorphic to $L^p(\mu_2, X),$ $VB^p(\mu_2, X)$, $cabc(\mu_2, X)$ respectively, if $L^1(\mu_1)$ is strongly isomorphic to $L^1(\mu_2)$.