TITLE: Does convergence in $L^p$ and pointwise imply the same limit?
QUESTION [8 upvotes]: If $f_n\in L^p$ converge to $f$ pointwise and to $g$ in $L^p$. Does that mean $f=g$ almost everywhere?

REPLY [4 votes]: By Fatou's lemma, we have $$\int \liminf_{n \to \infty} |f_n-f|^p d\mu \leq \liminf_{n \to \infty} \int |f_n-f|^p d\mu.$$
Since $f_n$ converges to g pointwisely, we have
$$\liminf_{n \to \infty} |f_n-f|^p=\lim_{n \to \infty} |f_n-f|^p=|g-f|^p.$$
Also, $f_n$ converges to f in $L_p$, then
$$\liminf_{n \to \infty} \int |f_n-f|^p d\mu=\lim_{n \to \infty} \int |f_n-f|^p d\mu=0.$$
Therefore, we have
$$\int |g-f|^p d\mu=0,$$
which implies that $g=f$ almost everywhere.