Weak convergence and pointwise convergence of norm

Question

Assume that $f_n\rightharpoonup f$ weakly in some $L^p$ space, and $|f_n|\rightarrow|f|$ pointwise. Does this imply that $f_n\rightarrow f$ pointwise? (or a subsequence?)

Answer 1

Consider $f_n(x)=-\mathbf{1}_{\mathbb{Q} \cap [0,1]}$ and $f(x)=\mathbf{1}_{\mathbb{Q} \cap [0,1]}$.

Then $|f_n(x)|=|f(x)|$ and $f_n \rightharpoonup f$ since

$$\forall \phi \in L^q, \int (f_n-f)\phi=0,$$

as $f_n-f$ is non-zero over a set of $0$ measure.