Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery.
I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given a Borel regular measure $ \mu $ in $\mathbb{R}^n $, given a $\mu$-measurable subset $E \subset \mathbb{R}^n $, let $$ \psi(x,E) = \lim_{r\rightarrow 0}\frac{\mu(E\cap B(x,r))}{\mu(B(x,r))} $$ Here $\psi $ is defined wherever the limit exists. Then do we have interior and exterior as $$ Int_\mu(E) = \{ x \in \mathbb{R}^n\ |\ \psi(x,E) = 1\},\\ Ext_\mu(E) = \{ x \in \mathbb{R}^n\ |\ \psi(x,E) = 0\} $$ and boundary $$\partial_\mu(E) = \mathbb{R}^n\setminus (Int_\mu(E)\cup Ext_\mu(E)) $$ I wold be extremely obliged if someone can tell me whether these are correct and whether they imply $$ \mu(E\setminus Int_\mu(E))= \mu(E^c\setminus Ext_\mu(E)) = \mu(\partial_\mu(E)) = 0 $$ from Lebesgue differentiation theorem. Also how exactly the topological and measure theoretic boundaries are related ? There is also another question, pardon me if it's too lame. Given a $\mu $ can we define a topology on $\mathbb{R}^n $ such that it's topological interior and boundary are the measure theoretic interior and boundary ?