I was wondering whether there is a formalized version of the following idea: We are given a function $f \in L^1(\mathbb R, \mu)$ for some absolutely continuous measure $\mu$ (but I'm also interested in generalization to more complicated spaces than $\mathbb R$). This function is of course defined only up to negligible sets, i.e. the term $f(x)$ does not make any sense outside of an integral. Can we uniquely extend this function to be pointwise defined by setting $$ \tilde f(x) = \lim_{n\to\infty} \frac{\int_{A^x_n}f(y)\mathrm d \mu(y)}{\mu(A_n^x)}$$ for open sets $A_n^x$ (i.e. $\mu(A_n^x) > 0$) such that $\limsup A_n^x = \{x\}$ (or maybe we constrain to monotonous sequences $A_{n+1}^x \subset A_n^x$, then we need $\bigcup_n A_n^x = \{x\}$)?
Then $\tilde f$ would be the "pointwise continuation" of $f$.
Obviously, the limit will always exist in some sense. The only trouble can originate in $\tilde f$ being ill-defined, i.e. depending on the sets $A_n^x$. But I am having difficulties constructing a function $f$ and sets $A_n^x$ such that this problem arises.