I am reading and studying one article in Tao's blog. Several quantitative versions of Lebesgue differentiation theorem are discussed there. I have gone through most of the proofs he left to the readers. However, I have difficulty understanding his hint in the last version of Lebesgue differentiation theorem. The statement is in the following:
If $\epsilon$ > 0, F: ${\Bbb N}\to {\Bbb N}$, and f: $[0,1] \to [0,1]$ is measurable, then there exists an positive integer n = $O_{\epsilon,F}(1)$ such that for all x in [0,1] outside of a set of measure $O(\epsilon)$ we have $|\frac{1}{r} \int_x^{x+r} f(y)\ dy - \frac{1}{s} \int_x^{x+s} f(y)\ dy| \leq \epsilon $ for all $2^{-n-F(n)} < r, s < 2^{-n}$.
My first attempt is to apply the strong regularity lemma to decompose $f=c+e+h$ on the dyadic intervals $I$ of length $2^{-n}$, where $c = \frac{1}{|I|} \int_I f(y) dy$, $e$ is small in the sense that $\frac{1}{|I|} \int_I |e(y)|\ dy \leq \epsilon$, and $h$ has vanishing averages in the sense that $\int_J h(y) dy = 0$ for all dyadic subintervals $J \subset I$ with $|J| \geq 2^{-F(n)} |I|$.
Tao then suggests we use Hardy-Littlewood maximal inequality to deal with the errors $e$. I still cannot see why it works here and how the theorem can be deduced fairly easily.