BEST APPROXIMATION IN L1

Let ( OMEGA , F, m) be a finite measure space and beta be a sub- sigma -algebra of F. If X is a Banach space, we let L**1( psi F, X), ((L**1( psi , beta , X)) denote the space of F-measurable ( beta -measurable) Bochner integrable functions with respect to m. It is proved that if X is reflexive and ( psi , beta , m) is separable, then L**1( psi , beta ,X) is proximinal in L**1( psi ,F,X).