Is a pointwise limit of $L^p$ functions in $L^p$ under certain conditions? Let $I=[0,1]$ with Lebesgue measure and $1\leq p<\infty$. If $f_k$ is a sequence in $L^p(I)$ with $\|f_k\|_p\leq 1$ for all $k$, and $f(x)=\lim_{k\rightarrow\infty} f_k(x)$ exists almost everywhere, must $f$ belong to $L^p(I)$?

By Fatou's lemma,
$$\int_0^1|f(x)|^p\;dx\leq \liminf_{n\to\infty}\int_0^1|f_n(x)|^p\;dx\leq 1$$
so $f$ is in $L^p$.
