Definition (Essential bound)

Let $(X,\mathscr{M},\mu)$ be a Measure Space and $f:X\to[0,+\infty]$ be measurable. We say that $M\ge 0$ is an essential bound for $f$ if and only if $$\mu(\{ x \in X:f(x)>M \})=0$$ If $f$ has an essential bound we say it is essentially bounded.

Definition ($L^{\infty}$)

$$L^{\infty}:=\{ f:X\to \mathbb{C}:f\text{ measurable and }\|f\|_{\infty}<\infty \}$$

Definition ($\infty$ norm)

For $f:X\to \mathbb{C}$ essential bounded we define the infinity Norm as $$\|f\|_{\infty}:=\inf\{ M\ge 0:M\text{ is an essential bound for }f \}$$