Measure and $L_\infty$ space

Question

Consider a function $f \in L_\infty$. I am trying to see if the following statement is true and if so why.

$$ \mu\{\, \vert f \vert = \Vert f \Vert_\infty \} \stackrel{??}{>} 0 \text { and } \{\, \vert f \vert = \Vert f \Vert_\infty \} \stackrel{??}{\neq} \emptyset$$

Any help is appreciated.

Answer 1

Consider the function $f(x)=x$ on $(0,1)$ and extend it by $0$ if you want a counterexample on $\mathbb{R}$. Then $\|f\|_\infty =1$. But $$ \{x\in(0,1)\;;\;|f(x)|=1\}=\emptyset. $$ So both statements are false.