I'm new to this forum. I don't have a formal education in mathematics and I'm trying to learn real analysis from Folland by myself. I'm not sure if I've understood the proof of theorem 3.20 correctly and your input would be appreciated.
Definition: $L_f = \{x: \lim_{r \rightarrow 0} \frac{1}{m(B(r,x))} \int_{B(r,x)} |f(y) - f(x)| dy = 0\}$.
Here is an image of the theorem and its proof. Theorem 3.20
The way I understand this proof is that $E^c \subset L_f$ and $E$ contains points belonging to both $L_f$ and $L_f^c$. But, since $m(E) = 0$ and $L_f^c \subset E$, $m(L_f^c) = 0$. Is this correct or am I missing something? All help is greatly appreciated. Thank you!