I got stuck in the definition of the essential support of $f$. I basically want to prove Hölder's inequality for the case $p=1$ and $q= \infty$. The proof is based on the fact that $$|g| \leq \|g\|_{\infty} \;\;\text{a.e.}$$ Now I don't see why this should be true by the definition of the essential support of $g$. This is given as follow: $$\|g\|_{\infty}:=\sup\{t >0 \,|\, \mu\{x \in E:|f(x)|>t\}>0\}$$ But why we have than the inequality $|g| \leq \|g\|_{\infty}$?
Many thanks for some help!
Indeed, $$\|f\|_\infty = \inf \{C \geq 0: |f(x) |\leq C \ \text{ for a.e. $x\in \Omega$}\}.\tag{$\star$}$$ You can easily prove that $\|f\|_\infty$ defined in eq. ($\star$) is equal to yours.
There exists a sequence $C_n\geq 0$ such that $C_n \to \|f\|_\infty$ and for each $n$, $|f(x)|\leq C_n$ a.e. on $\Omega$. Therefore $|f(x)|\leq C_n$ for all $x\in \Omega\setminus E_n$, with $\mu(E_n) =0$. We set $E= \bigcup_{n=1}^\infty E_n$, so that $\mu(E)= 0$ and $$|f(x)|\leq C_n\ \ \ \forall n, \quad \forall x\in \Omega \setminus E;$$ it follows that $|f(x)|\leq \|f\|_\infty$, $\forall x\in \Omega\setminus E$.