Essential supremum and bounded funtions

Question

For a funtion $f\in M(X,\mathcal{A})$(bounded funtions) we have $f\leq $ ess sup$|f|$ , My question is why acully we need bouned funtions ,can you give me an example of an unbounded funtion for the statement is not true?

Answer 1

For any measurable function $f$ we do have $|f| \leq esssup |f|$ but usually one defines essential sup only for essentially bounded functions.There is no harm in defining $\|f\|_{\infty}$ to be $\infty$ when there is no $M$ such that $|f| \leq M$ a.e. so the inequality holds for any measurable $f$.