Let $X$ be compact metric space, and $\mu_n$ a sequence of finite Borel measures on $X$ such that $\sup_n \mu_n(X)<\infty$. Prove
(a) For every $f \in C(X)$, there exists a subsequence $n_j$ such that $\int f d\mu_{n_j}$ converges.
(b) Let $B\subset C(X)$ be countable dense, show there exists a subsequence $n_j$ such that $\int f d \mu_{n_j}$ converges for all $f \in B$.
(c) let $n_j$ be as in the previous, prove $\int f d\mu_{n_j}$ converges for all $f \in C(X)$.
(d) Let $L:C(X) \to \Bbb{R}$ be given via $L(f):=\lim_{j \to \infty} \int f d\mu_{n_j}$. Show $L$ is a positive linear functional.
Attempt:
I have prove (a) and have so whats the difference between (a) and (c), and for (b) do I use the fact that $\bar{B}=C(X)$? Or for (b) can I say let $B=\{f_n\}_{n \in \Bbb{N}}$ then since $f_n:X \to \Bbb{R}$ and $f$ is continuous and $X$ is compact there exists $a,b \in \Bbb{R}$ such that $a \leq f_n(x) \leq b$ thus $\int f_n d \mu_{n}$ is bounded beflow by $aA$ and above by $bA$ thus its a bounded sequcne of reals thus has convergent subsequence. And for (c) do I use (a) or?
For (d), I need to show for $f,g \in C(X)$ that $L(f+g)=L(f)+L(g)$, i.e.,
$$\lim_{j \to \infty}\int (f+g) d \mu_{n_j}=\lim_{j \to \infty}\int f d \mu_{n_j}+\lim_{j \to \infty} \int g d \mu_{n_j}$$
which if my functions were non-negative I could invoke linearity? Or do I need some convergence theorem to swap limit for integral? And if $f,g$ converges to different things theyre the same since were in a Hausdorff space, sorry i guess what ime trying to say is if the limit of integral of sums is the sum of the limits of the integrals.