Let $X$ be a compact metric space equipped with the Borel $\sigma$-algebra
Then we have $C(X)$, the set of all the real-valued continuous maps on $X$, equipped with the sup-norm.
We may also define $BM(X)$ as the set of all the real valued bounded measurable functions on $X$, and equip this too with the sup norm.
Clearly, we have $C(X)$ sitting inside $BM(X)$.
Question. Is $C(X)$ dense in $BM(X)$?
I guess the question boils down to asking that if $E$ is a Borel set in $X$ then there is a sequence of continuous functions $(f_n)$ such that $\|\chi_E-f_n\|_\infty\to 0$. But is this true?