Let X be a Banach space with dual X ∗ {X^\ast } and let ( Ω , Σ , μ ) (\Omega ,\Sigma ,\mu ) be a finite measure space. Suppose f : Ω → X f:\Omega \to X is weakly measurable. There exists a (norm) measurable g : Ω → X g:\Omega \to X such that ⟨ x ∗ , f ⟩ = ⟨ x ∗ , g ⟩ \langle {x^\ast },f\rangle = \langle {x^\ast },g\rangle a.e. for each x ∗ ∈ X ∗ {x^\ast } \in {X^\ast } if and only if each set A of positive μ \mu -measure has a subset B of positive μ \mu -measure such that there is a weakly compact convex subset K of X with the property that \[ ⟨ x ∗ , f ⟩ ⩽ sup x ∈ K ⟨ x ∗ , x ⟩ \langle {x^\ast },f\rangle \leqslant \sup \limits _{x \in K} \;\langle {x^\ast },x\rangle \] μ \mu -almost everywhere on B for each x ∗ ∈ X ∗ {x^\ast } \in {X^\ast } .