Considering $L^p$ $( 1 \leq p < \infty)$ as a normed vector space, each element of $L^p$ is actually an Equivalent class. Take $[f] \in L^p $ as an Equivalent class, What is the Nicest possible function $g$ such that $g \in [f]$ (i.e. $g=f$ almost everywhere).
By the word Nice you are free to consider any good topological or algebraic property like continuity, differentiability, boundedness, etc... (as many as you can)
Second question: Let $N$ denotes the set of such Nice Properties, imposing $N$ as set of conditions on $g$, can one identify $g$ uniquely ? In other words can we rewrite $L^p$ in following way
$$L^p = \{ g : R\rightarrow R \cup \{\pm \infty\} \quad | \quad \|g\|_p < \infty ,~g\text{ satisfies }N \}$$
This makes we can think about $L^p$ as a set of nice functions, free of any confusing equivalent classes.
Clarification: My intention is to find a set of finite conditions called $N$ , when we imposing them, $g$ is determined uniquely, in order to replace this $g$ by $[f]$ to rewrite $L^p$ as above. Lusin's theorem says one can pick $g \in [f]$ such that $g$ is continuous except on very small set (as small as you want but might have positive measure).This might be helpful but still can't give us the set of condition $N$ !
