In my text book all it said about $L^\infty(\Omega)$ space is that it is the space of all measurable functions that are bounded almost everywhere. No example given.
I can't see how any member of this space would be useful. Perhaps $f = e^{-ax}$ would be a member of $L^\infty(\Omega)$ space? But it is more useful in a $L^2(\Omega)$ context i.e. signal analysis.
Can someone provide some motivating example of common functions $f$ such that $f \subset L^\infty(\Omega)$?