Given a finite measure on a compact, take $f_n\in L^1$ with norms $\leq 1$ and suppose that $\int f_n g$ tends to a limit for all continuous $g$. Is it true that then $\int f_n g$ converge for any $g\in L^\infty$? How can one prove this?
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If $(f_n)$ is a bounded sequence in $L^1(X,\mu)$, it is true that the set of $g\in L^\infty(X,\mu)$ such that $\int_X f_n g\, d\mu$ has a limit in $\mathbb{C}$ is a normclosed linear subspace of $L^\infty(X,\mu)$. Thus, from a dense set of test functions $g$ you can infer the weak convergence of the sequence $(f_n)$ (to an element of ($L^\infty)^*$, of course, in general not in $L^1$). In conclusion, in general continuous functions are not enough (they are a closed subspace, usually proper, of $L^\infty$), but e.g. simple functions, which are uniformly dense, will do. 

