"Sets of finite perimeter and geometric measure theory" F.Maggi, page 264 in the middle of the page ^^' Maybe i formulated it wrong.. It is strange that the book made an error like this

Thank you very much! For convenience of future readers i add that the same argument that you presented works with the compactness of the sets of finite perimeter: that is If $\{ E_h\}_h$ are of finite perimeter and they satisfies $sup_h P(E_h)<\infty$ and $E_h\subset B_R$ for a certain R Then there exists $E$ a set of finite perimeter s.t. $E_h\to E$ and $\mu_{E_h}\to^{\star}\mu_E$.