Given the following definitions: $$\mathcal{L}^{\infty} = \{ \ f\in\mathcal{L}^0 \ |\ \exists a\in\mathbb{R}_+: |f| \leq a \ \ \text{a.e.}\}$$ $$\|f\|_{\infty} = \inf_{|f| \leq a \ \ \text{a.e.}} a$$
I'm trying to prove the following property:
If $f\in\mathcal{L}^{\infty}$, then $|f|\leq\|f\|_{\infty}$ a.e.
In particular, I'm having to trouble proving that the set $\{x: |f(x)| > \|f\|_{\infty} +\frac{1}{n}$} has zero measure. What is the procedure to prove this?
by definition of "inf", there exsit some $a_n$ satisfies $\{x:|f(x)|> a_n \}$ has measure zero and $\|f\|_{\infty} \le a_n<\|f\|_{\infty} +\frac{1}{n}$, for each $n$.
But we know that $$\{x: |f(x)| > \|f\|_{\infty} +\frac{1}{n}\}\subset \{x: |f(x)| > a_n\}$$ which implies that $\{x: |f(x)| > \|f\|_{\infty} +\frac{1}{n}\}$ is also measure zero.