ON THE PETTIS INTEGRABILITY

A function f : Ω → x is called intrinsically-separable valued if there exists E ∈ Σ with μ(E) = 0 such that f(Ω - E) is a separable in X. For a given Dunford integrable function f : Ω → X and a weakly compact operator T, we show that if f is intrinsically-separable valued, then f is Pettis integrable, and if there exists a sequence (fn) of Dunford integrable and intrinsically-separable valued functions from Ω into X such that for each x* ∈ X*, x* fn → x* f a.e., then f is Pettis integrable. We show that a function f is Pettis integrable if and only if for each E ∈ Σ, F(E) is weak*-continuous on Bx* if and only if for each E ∈ Σ, M = {x* ∈ X* : F(E)(x*) = 0} is weak*-closed.