Take the 2-minute tour ×  Can anything be said about the measure
of the topological boundary of a Cacciopoli set in $R^n$? Of
course, the reduced boundary has finite (n-1)-dimensional
Hausdorff measure, but this does not say anything about the
topological boundary, for instance, points with density 0 or 1
can still be part of the boundary.  My precise question: if $E$
is a Caccioppoli set, does there exist a measurable Cacciopoli
set $F$, such that $E\triangle F$ and $\partial F$ are both
Lebesgue null sets?  share|improve this question  1 Answer 1  up
vote 4 down vote accepted  The answer is no. Take countably many
disjoint closed balls $B_i$ contained in the square
$Q=[0,1]\times [0,1]$ and such that: (i) Sum of areas of $B_i$
is less than 1 (ii) Sum of perimeters of $B_i$ is finite (iii)
$\bigcup B_i$ is dense in $Q$ Since the series $\sum \chi_{B_i}$
converges in BV norm, the set $E=Q\setminus \bigcup B_i$ has
finite perimeter. It also has positive measure and empty
interior. Any representative $F$ of the