 Solution

For $1\le p<\infty$, $L^p(E,\mathcal{E},\mu )$ consists of measurable functions with ${\int }_E|f{|}^{p}d\mu <\infty$, identifying functions equal almost everywhere. Its Lp norm is $\Vert f{\Vert }_p=(\int |f{|}^p{)}^{1/p}$. Thus membership is a condition on an almost-everywhere equivalence class, not on individual point values.