An almost everywhere limit of functions bounded in $L^p$ is bounded in $L^p$

If $\norm{f_k}_p\le M$ for all $k$ and $f_k\to f$ almost everywhere, then $\norm{f}_p\le M$.Suggest a slogan

Let $(X,\mcm,\mu)$ be a measure space, $1\leq p\leq\infty$, $M\geq0$, and let $f_k\colon X\to\CC$ be measurable with $\norm{f_k}_p\leq M$ for all $k\geq1$. If $f_k\to f$ $\mu$-almost everywhere for a measurable $f\colon X\to\CC$, then $f\in L^p(\mu)$ and $\norm{f}_p\leq M$.