same book, Example 12.25, there you have an open set of finite perimeter E in R2 with

|spt mu_E|>0. Since H^{n-1}(pa^* E) is finite you have

|pa^* E|=0 thus |(spt mu_E)\ pa^*E|>0 thus H^{n-1}((spt mu_E)\ pa^*E)= + infty

In Maggi's book, Example 12.25, there you have an open set of finite perimeter $E$ in $\mathbb{R}^2$ with $|\text{spt}\, \mu_E|>0$. Since $H^{n-1}(\partial^* E)$ is finite, you have $|\partial^* E|=0$. Thus $|(\text{spt}\, \mu_E)\setminus \partial^*E|>0$, so $H^{n-1}((\text{spt}\, \mu_E)\setminus \partial^*E)= + \infty$