Coproximinality in Spaces of Bochner Integrable Functions

In this short note we use a new way of applying von Neumann's selection theorem for obtaining best coapproximation in spaces of measurable functions. For a coproximinal closed subspace $Y$ of a Banach space $X$, we show that if $Y$ is constrained in a weakly compactly generated dual space, then the space $L^1(\mu,Y)$ of $Y$-valued Bochner integrable functions is coproximinal in $L^1(\mu,X)$. This extends a result of M. R. Haddadi, N. Hejazjpoor and H. Mazaheri [{\it Some results about best coapproximation in $L^P(S,X)$}, Anal. Theory Appl. 26 (2010) 69--75], proved when $Y$ is reflexive.