Approximative compactness in B\"ochner spaces

For any p∈[1,∞), we prove that the set of simple functions taking at most k different values is proximinal in B\"ochner spaces Lᵖ(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∈(1,∞) and even approximatively norm-compact under stronger hypothesis.