On Various Modes of Scalar Convergence in $L_0 ( \mathfrak X )$
S. J. Dilworth and Maria Girardi
A sequence $\{f_n\}$ of strongly-measurable functions taking values in a Banach space ${\mathfrak X}$ 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 $x^*\in {\mathfrak X}^*$. Let $1\le p\le \infty$. The main questions addressed in this paper are whether an $L_p({\mathfrak X})$-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 $L_p({\mathfrak X})$-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 often depend upon $p$ and upon the geometry of ${\mathfrak X}$.