TITLE: Elements of $L^p$ and nice representatives of equivalence classes
QUESTION [8 upvotes]: 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 $N$ !

REPLY [2 votes]: In harmonic analysis one frequently defines a modification of $f$ by taking the new value $f(x)$ as the limiti of averages of $f$ over balls centered at $x$, with radius tending to 0. You do this at every point for which the limit exists. If $f$ is equivalent to a continuous function, this method produces the continuous representative. By Lebesgue differentiation this limit exists at almost every $x$ and coincides with the original value of $f(x)$ for any locally $L^1$ function; of course it remains the question of how to redefine $f$ at points where the limit do not exist; maybe one can set $f=0$ there. I guess you can not do better than this, but I might be wrong since I do not know lifting theory.