Assume that $f \in L^1_{loc}(R^n)$ and let $x\in R^n$ and let $x\in R^n$. Assume that for every sequence $E_1, E_2 , E_3\ldots$ converging regularly to x there exists
$\lim_{k\to\infty} \frac{1}{\lambda(E_k)} \int_{E_k} f(y) dy $.
Prove that x belongs to the Lebesgue set of f .
I guess that I will have to begin by showing that I can assume that f is real valued, and then show that the above limit obtained by interlacing two sequences is independent of the particular sequence in $E_k$ and then probably proceed !!
Atleast this is the brief outline that I discussed with a math PHD student , but I cannot see how to proceed formally in this way.
Definition of lebegue set :
Suppose $ f \in L^1_{loc} and , X \in R^n$ . Then x is a point in the lebesgue set of f if there exists a number A such that
$lim_{r=0} 1/(\lambda(B(x,r)) * \int_{B(x,r)} |f(y)-A|dy = 0 $