It’s on the book of Sets of Finite Perimeter and Geometric Variation Problems by Francesco, on Page 110.
My questions are:
(12.19) Why here use the closure of $A$ rather than $A$?
The definition of $A_k$, why $dist(x, \partial A) < 1/k$, rather than $>$?I think it should be limit to $A$.
If it should be used $>$ here, why not use $A_k = A \cap B_k$?