Weak* convergence on $L^\infty(\Omega)$ and almost everywhere convergence

Question

Let $\Omega$ be finite measure space. Suppose that $f_n \to f$ in $L^\infty(\Omega)$ for the weak* topology.

Does there exists a subsequence (or a subnet) $(f_{n_k})$ such that $f_{n_k} \to f$ almost everywhere?

Same question with the additional assumption $\|f_n\|_{L^\infty} \leq C$ for some constant $C$.

Answer 1

Does weak convergence in $L^2$ implies convergence almost everywhere along subsequence?