Approximative compactness in Böchner spaces

For any $p\in [1,\mathrm{\infty })$, we prove that the set of simple functions taking at most $k$ different values is proximinal in Böchner spaces $L^p(X)$ whenever $X$ is a dual Banach space with $w^{\ast }$-sequentially compact unit ball. With additional properties on $X$ and its norm, we show these sets are approximatively $w^{\ast }$-compact for $p\in (1,\mathrm{\infty })$ and even approximatively norm-compact under stronger hypothesis.