  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.
