$L^p$ norms approach $\norm{f}_\infty$ on finite measure spaces

On a finite measure space, $\norm{f}_p\to\norm{f}_\infty$ as $p\to\infty$.Suggest a slogan

Let $(X,\mcm,\mu)$ be a measure space with $\mu(X)<\infty$, and let $f\colon X\to\CC$ be measurable. Then, in $[0,\infty]$, \[ \lim_{p\to\infty} \norm{f}_p = \norm{f}_\infty , \] where $\norm{f}_\infty$ is the essential supremum of $\abs{f}$.