This question is related to Question 2 of [my previous posting](https://mathoverflow.net/questions/103860/the-monotone-closure-of-a-c-algebra).

> **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?
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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.



Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?
No. The bounded Baire class one functions on $[0,1]$ are stable under uniform limits and hence constitute a C\*-algebra. This C\*-algebra contains every semicontinuous function on [0,1]. Every function in $L^\infty[0,1]$ is equal almost everywhere to a Baire class two function, but not a Baire class one function. (The previous version of my answer neglected this essential point.)


See <http://www.encyclopediaofmath.org/index.php/Baire_classes>


