Functions in $L^p$ are only defined $µ$-almost everywhere, so for a given evaluation point $x$, $F(x)$, $f\in L^p$ can be changed to any value, so in general it would not be well-definied to just write $y=f(x)$. (Every representant $f$ of the equivalence class $F$ can be evaluated, okay. But this doesn't help.)
What is needed to have a well-defined evaluation mapping $(F,x) \mapsto F(x)$ , and how is this problem solved canonically? It would be enough for example that F is piecewise continuous, then I could define the evualation at every continuous part. But isn't much less (for example piecewise one-sided continuity?) enough?
And if you take for example Sobolev spaces, and consider the weak derivative of $F:x \mapsto |x|$. Does it follow from the definition of weak derivative in Sobolev spaces that we choose the continuous representant of $F'$ in $L^p$, and define the evaluation of the weak derivative only on the continuous part? How is this problem handled?
Canonical choices for values of measurable functions ... see the theory called "lifting"
For example:
Topics in the Theory of Lifting
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, 1969)
Alexandra Ionescu Tulcea, C. Ionescu Tulcea