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

Suppose $(X,\mathrm{\Sigma },\mu )$ 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.