Existence of best simultaneous approximations in Lp(S,Σ,X)Lp(S,Σ,X)

Let (S,Σ,μ)(S,Σ,μ) be a complete positive σσ-finite measure space and let XX be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in Lp(S,Σ,X)Lp(S,Σ,X). Let Σ0Σ0 be a sub-σσ-algebra of ΣΣ and YY a nonempty locally weakly compact convex subset of XX such that spanY¯ and its dual have the Radon–Nikodym property. We prove that Lp(S,Σ0,Y)Lp(S,Σ0,Y) is NN-simultaneous proximinal in Lp(S,Σ,X)Lp(S,Σ,X) (with the additional assumption that (S,Σ,μ)(S,Σ,μ) be finite for the case when p=1p=1). Furthermore, for the special case when Σ0=ΣΣ0=Σ, we show that the assumption that the dual of spanY¯ has the Radon–Nikodym property can be removed.