My definition for the essential supremum norm as follows. For measurable $f$ defined on a measurable set $E$, the essential supremum norm of $f$ over $E$ is $$ ||f||_{L^{\infty}(E)} = ||f||_{\infty} = \inf \left\{\sup_{x \in A} |f(x)|: A \text{ measurable}, m(E \setminus A) = 0\right\}. $$
I was reading a proof showing that $||f||_{\infty} = 0 \implies f = 0$ almost everywhere, which used the fact that
$$ ||f||_{\infty} = \inf \left\{ \alpha \in \mathbb{R}: m\left\{x: \left|f(x)\right| > \alpha\right\} = 0\right\}. $$
I am struggling to prove this alternate form of $||f||_{\infty}$ from my definitions of measurable functions, sets, etc. though, and would appreciate some help.