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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?

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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

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    In $L^p$ there is certainly a linear lifting, but it is just that there is none that preserves products (as we require in $L^\infty$).2012-07-25