On Various Modes of Scalar Convergence in L0

A sequence fn of strongly measurable functions taking values in a Banach space is scalarly null a.e. (resp. scalarly null in measure) if x∗fn → 0 a.e. (resp. x∗fn → 0 in measure) for every x∗ ∈ ∗. Let 1 ≤ p ≤ ∞. The main questions addressed in this paper are whether an LP -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 Lp -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 .