In Federer's Theorem, $ H^{n-1} (\partial ^{m}E - \partial ^{*}E)=0 $, where $E$ is a set of finite perimeter in $ \mathbb R^n $, $\partial ^{m}E$ is the measure theoretical boundary of E, and $\partial ^{*}E$ is the reduced boundary of E.

From Maggi's book Prop. 12.19, We know that $sptE = \{x \in \mathbb R^n : 0<|E \cap B(x,r)|< \omega _ n r^n, \forall \space r > 0\}$, and by definition, $\partial ^{m}E \subseteq sptE$

Now here comes my question, is it true that $ H^{n-1} (sptE - \partial ^{*}E)=0 ?$

I thought about this question for several days, but I could not prove it nor could I give a counterexample. I'm lack of some pictures in mind.

Also, I'm curious about this: We know that the topological perimeter of a Koch Snowflake is infinite. But in geometric measure theory, we study the distributional perimeter. How to show that the distributional perimeter of the Koch Snowflake is infinite. (This might be a dumb question, but it bothered me for a while.)

Any idea would be really appreciated.