I have stubbornly been attempting to prove that every Lebesgue integrable function $f$ has Lebesgue points almost everywhere, i.e., satisfies $$ \lim_{r\rightarrow0}\frac{1}{m(B_r)}\int_{B(x,r)}|f(y)-f(x)|\ dy=0 $$ for almost all $x\in\mathbb{R}^k$, by my own means. Every proof of this result I have found utilizes the Hardy-Littlewood Maximal function, and the related inequality.
My question is this: Is there some proof of the above result that doesn't rely on the Maximal function, preferably one with comparable brevity to the proof given in, for instance, Rudin's RCA?