I'm trying to answer the following question, however, I'm stuck on some parts and would appreciate some help.
Let $X$ be a compact metric space and suppose $\mathcal{F} \subseteq \{ f \in \mathbb{C}^X \mid f \, \text{is bounded and Borel measurable} \, \}.$ Furthermore, suppose $\mathcal{F}$ is a complex vector space containing $C(X, \mathbb{R})$ ( the set of all continuous, real valued functions on $X)$, and that for any uniformly bounded sequence $(f_n)_{n \in \mathbb{N}} \in \mathcal{F}^{\mathbb{N}}$, if $f_n$ converges to some function $f \in \mathbb{C}^X$ pointwise, then $f \in \mathcal{F}$.
Then, $\mathcal{F}$ consists of all bounded, Borel-measurable functions on $X.$
I'm trying to prove this in three parts. First, we show that the set $L_0$ given by
$$ L_0 = \{ E \subseteq X \mid E \text{ is measurable and } 1_E \text{ is a pointwise limit of a sequence of uniformly bounded continuous functions} \}.$$
is an algebra containing all the open sets in $X$. (Note: $1_E$ is the characteristic function on $E$ ).
To see this, clearly $X$ is measurable since it is open, and we can take the sequence of functions $f_n : X \rightarrow \mathbb{C}$, $f_n(x) = 1$. Then this satisfies the criteria. However, I'm not sure how to show that if $A, B \in L_0$, we have both $A \cup B \in L_0$ and $A \setminus B \in L_0$. I've tried defining a function $$ g_n(x) = \begin{cases} a_n(x) + b_n(x) & x \in A \triangle B \\ a_n(x)b_n(x) & x \in A \cap B \\ 0 & x \in (A \cap B)^c \\ \end{cases} $$ where $a_n \rightarrow 1_A$ and $b_n \rightarrow 1_B$ however I don't think this will satisfy the continuity criteria.
For the next step I have to show that the following set $L_1$ is a monotone class (https://en.wikipedia.org/wiki/Monotone_class_theorem),
$$L_1 = \{ E \subseteq X \mid E \text{ is measurable and } 1_E \in \mathcal{F} \}. $$
Once this is done, I should use the monotone class lemma to conlcude $L_1$ contains all the Borel sets in $X$. This part is obvious, as $L_0 \subseteq L_1$ and if we showed that $L_0$ contains the open sets earlier, then in turn $L_1$ contains the Borel $\sigma$-algebra generated by the open sets.
The last bit is then to conclude that $\mathcal{F} $ contains all bounded, Borel measurable functions on $X$.