Approximative compactness in Böchner spaces

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