On the lattice structure of the space of all Bochner integrable Banach lattice-valued functions

TL;DR

This paper investigates the lattice structure of the space of Bochner integrable Banach lattice-valued functions, establishing conditions under which it inherits properties like being a KB-space or having the Fatou property from the target space E.

Contribution

It characterizes when the space of Bochner integrable functions inherits lattice properties from the Banach lattice E, providing new insights into their order structure and convergence behavior.

Findings

B(X,E,μ) is a KB-space iff E is a KB-space.

B(X,E,μ) has the sequential Fatou property iff E does.

Results on Bochner integral convergence using E's order structure.

Abstract

Suppose $(X,\Sigma,\mu)$ is a finite measure space, $E$ is a Banach lattice, and $B(X,E,\mu)$ is the space of all Bochner integrable $E$-valued functions. In this note, we show that $B(X,E,\mu)$ is a $KB$-space or has the sequential Fatou property if and only if so is $E$. Among this, some results about Bochner integral convergence in $B(X,E,\mu)$, using order structure of $E$, have been proved, as well.