Let \((\Omega, \mathcal{A}, \mu)\) be a measure space and \(f \in L^\infty(\Omega, \mu)\) . Is it true that a point \(x \in \Omega\) exists with
\begin{equation*} |f(x)|\ge \lVert f\rVert_{L^\infty(\Omega)}? \end{equation*}Do we need further assumptions for \(\mu\)? Should it be finite as in 0x669e0e75 ?
Remarks
- Indeed it is possible, that a Supremum is never reached. For example one could consider \begin{equation*} f(x) = 1 - \exp(-x) \end{equation*} for \(x\ge 0\). Then clearly no point \(x\ge 0\) exists, such that \(f(x)\ge 1\).