This question is related to Question 2 of my previous posting.

It suffices to consider whether the indicator function of any Borel set is obtained from $S$ by algebraic operations and (essential-supremum) norm limits. If necessary, you may assume $\Omega$ to be second countable. Thank you.

Question.Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded Borel measurable functions on $\Omega$. Suppose that $S$ is the set of bounded lower or upper semi-continuous functions on $\Omega$. Does $S$ generate $L^{\infty}(\Omega,\mu)$ as a $C^*$-algebra?