Given $c>0$, can we produce sequence of measurable sets $E_n\subset\mathbb{R}$ such that their characteristic functions $\chi_{E_n}$ converge weakly in $L^p(\mathbb{R})$ to some $c\chi_A$ for some $A\subset\mathbb{R}$ with $\chi_A\neq0$?
https://math.stackexchange.com/questions/4086368/the-lp-limit-of-characteristic-functions-is-a-characteristic-function
The case $c=1$ is trivial because we can pick constant sequences.
The post Weak convergence of a sequence of characteristic functions implies that we can do it to $p=2$ and $c=\frac{1}{2}$.
The case $c>1,1\leq p\leq \infty$ is impossible. Since $\chi_A\in L^q(\mathbb{R})$ for all $1\leq q\leq\infty$, the condition $\chi_{E_n}\rightharpoonup 2\chi_A$ implies that $$|E_n\cap A|=(\chi_{E_n},\chi_A)\rightarrow(c\chi_A,\chi_A)=c\cdot|A|$$ which is absurd when $c>1$. Here one recalls $g\mapsto\int_{\mathbb{R}}g\chi_A\text{d}x$ is well-defined in $(L^p(\mathbb{R}))^*$.
How about other cases?