In [1] J. Alexopoulos has shown that if Φ ∊ δ2 and if its complement ψ satisfies limt ψ∞ψ(ct) ψ(t) = ∞ for some c > 0 then a bounded set K ⊂ L Φ is relatively weakly compact if and only if K has equi-absolutely continuous norms. Even though all such Φ fail the2 condition we do not know whether the theorem is applicable to all Φ ∊ δ2\2. In this paper we make some progress towards a generalization of this theorem. In particular we show that an N-function Φ/∊2 if and only if every weakly null sequence (c n χEn) in E Φ (μ) has equi-absolutely continuous norms. Other characterizations of the δ2 condition are given.