Let $(\Omega, \mu)$ be a semifinite measure space. I know that for any $1 < p \leq \infty$, it holds $$ \|f\|_p = \sup\left\{\left|\int fg \ {\rm d}\mu\right| \ : \ \|g\|_{p^\prime} \leq 1, \ g \ \mathrm{simple \ function}\right\}, $$ where $1/p + 1/p^\prime = 1$ since simple functions are $L^{p^\prime}-$norm-dense in $L^{p^\prime}$.
Simple functions are only weak$^*$-dense in $L^\infty$, so I would need the intersection of simple functions and the closed unit ball in $L^\infty$ to be weak*-dense in the closed unit ball of $L^\infty$.
Is this last statement true?