By assumption, for $p \in (1,\infty)$, I have a bounded sequence of functions $f_n$ in $L^p$ (that is, $L^p$ norms of the functions are uniformly bounded) that converges almost everywhere to a function $f$.
With this, by boundedness and Fatou's lemma, one deduces $f\in L^p$. Additionally, since $L^p$ is a reflexive Banach space for our choice of $p$, we know that there exists a weakly convergent subsequence $f_{n_k} \xrightarrow{w} g$ in $L^p$, some $g\in L^p$. If I am able to show $f=g$ almost everywhere, is this enough to conclude that $f_n$ converges weakly to $f$ in $L^p$?
