De Giorgi's structure theorem for reduced boundary - why compact submanifolds?
Solution 1:
I worked it out. The theorem only requires a countable union of compact sets. We can take a sequence of compact subsets of the form $K_h:=\{x \in \partial E : \operatorname{dist}(x,\partial E \backslash \partial^* E)\ge\frac{1}{h}\}$.