Hello I've been given the following question and I'm a bit confused.
Let $\mu$ be a Borel measure on X a metric space. Prove that the bounded continuous functions are separable in $L^p(\mu)$.
Now does this question only want us to consider continuous bounded functions that lie in $L^p(\mu)$ and use the $L^p$ measure, in which case I have a solution since simple functions are dense in $L^p$. However I'm not sure if the question intends for me to look at the space of continuous bounded functions including the functions that do not lie in $L^p$ and show that it is separable under the $L^P$ norm.
In the latter case I am unable to find either a proof or a counterexample.
Many Thanks