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 your 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
$$L^p = \lbrace g \colon \mathbb{R}\to \mathbb{R} \cup \lbrace \pm \infty\rbrace \mid \|g\|_P < \infty,~g \text{ satisfies }~N \rbrace$$
This makes we can think about $L^p$ as a set of nice functions, free of any confusing equivalent classes.