Approximation properties for spaces of Bochner integrable functions

For a finite measure space (Ω,A,μ), for a sub-σ-algebra B⊂A, and for a dual space X⁎, having the Radon-Nikodým property, we show that every A measurable X⁎-valued, Bochner integrable function has a best approximation in L1(B,X⁎). This extends a result of Papageorgiou, Shintani and Ando. For Banach spaces X, for which L1(A,X) is an L-embedded space, we obtain a complete analogue of the main results of Shintani, Ando and Papageorgiou for increasing sequence of sub-σ-algebras.