Approximation properties for spaces of Bochner integrable functions

For a finite measure space (Omega, A, mu), for a sub-sigma-algebra B subset of A, and for a dual space X*, having the Radon-Nikodym property, we show that every A measurable X*-valued, Bochner integrable function has a best approximation in L-1(B, X*). This extends a result of Papageorgiou, Shintani and Ando. For Banach spaces X, for which L-1(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-sigma-algebras. (C) 2014 Elsevier Inc. All rights reserved.