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 $, $\partial ^{e}E$ is the essential boundary of E, and $\partial ^{*}E$ is the reduced boundary of E.

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

Now here comes my question, is it true that $ H^{n-1} (spt \mu _E - \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.

Any idea would be really appreciated.

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

From Maggi's book Prop. 12.19, We know that $\operatorname{spt}(\mu _E) = \{x \in \mathbb R^n \mid 0<|E \cap B(x,r)|< \omega_n r^n, \forall \ r > 0\}$, and by definition, $\partial ^{e}E \subseteq \operatorname{spt} \mu _E$

Now here comes my question, is it true that $ \mathcal{H}^{n-1} (\operatorname{spt} \mu _E \setminus \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.

Any idea would be really appreciated.

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 $spt(\mu _E) = \{x \in \mathbb R^n : 0<|E \cap B(x,r)|< \omega _ n r^n, \forall \space r > 0\}$, and by definition, $\partial ^{m}E \subseteq spt \mu _E$

Now here comes my question, is it true that $ H^{n-1} (spt \mu _E - \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.

Any idea would be really appreciated.

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 $, $\partial ^{e}E$ is the essential boundary of E, and $\partial ^{*}E$ is the reduced boundary of E.

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

Now here comes my question, is it true that $ H^{n-1} (spt \mu _E - \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.

Any idea would be really appreciated.