On Various Modes of Scalar Convergence in L₀(X)

A sequence $\{f_n\}$ of strongly-measurable functions taking values in a Banach space $\X$ is scalarly null a\.e\. (resp. scalarly null in measure) if $x^*f_n \rightarrow0$ a\.e\. (resp. $x^*f_n \rightarrow 0$ in measure) for every $x^*\in \X^*$. Let $1\le p\le \infty$. The main questions addressed in this paper are whether an $L_p(\X)$-bounded sequence that is scalarly null a\.e\. will converge weakly a\.e\. (or have a subsequence which converges weakly a\.e\.), and whether an $L_p(\X)$-bounded sequence that is scalarly null in measure will have a subsequence that is scalarly null a\.e. The answers to these and other similar questions depend upon $p$ and upon the geometry of $\X$.