Let $1\leq p\leq\infty $ and let $T:L^p[0,1]\to L^p[0,1]$ be a linear operator. Suppose $T (f_n)$ converges to $T (f)$ almost everywhere whenever $(f_n)$ converges to $f$ almost everywhere. I want to show that $T$ is bounded.
I tried to do it like this with the help of Closed Graph Theorem.
Let $(f_n,T (f_n))$ be a sequence in graph of $T$ such that $(f_n,T (f_n))$ converges to $(f,g)$. Then $f_n\to f$ in $L^p [0,1]$ and $T (f_n)\to g$ in $L^p [0,1] $. Thus there exists a subsequence $(f_{n_k})$ converges a.e. to $f $. Now how to show $g=T (f)$?