Suppose $X$ is compact metric space. Let $A$ be the smallest set of complex functions containing all continuous functions such that:
If $f_n \in A$ are uniformly bounded and $f_n \to f$ pointwise then $f \in A$.
Is it true that $A$ is equal to the set of all Borel functions? If not, what is relation between these two? What assumptions are needed to get this?
Answers as well as references will be appreciated.