Suppose $(z_n)$ is a weakly convergent sequence in $L^p(\Omega)$, $\Omega$ is a limited open set of $\mathbb{R}^N$. Let $(\chi_n)$ be defined by $\chi_n(x)=\chi(x.n)$, where $\chi$ is the characteristic function of a rectangular block $B$. So, we can conclude that $\chi_n\stackrel{*}{\rightharpoonup}<\chi>_B$, where $<\chi>_B$ denotes the average (here the convergence is in $L^\infty(\Omega)$).
Is it possible to prove that $z_n \chi_n\rightharpoonup z<\chi>_B$? What kind of changes should I do to prove it?
Of course, I can't suppose strong convergence here.