Definition

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.