Lebesgue-Measure of special subsets I got two closed subsets $A,B\in\mathbb{R}^{m\times n} (\mathbb{R}^{n\cdot m}\text{ as vectorspace})$ , which satisfy to following properties:
$A\cup B=\mathbb{R}^{m\times n}$
$\partial A=\partial B$
$A^c = B\backslash{\partial B},B^c = A\backslash{\partial B}$
where $\partial A$ means the boundary and $A^c$ the complement of a set $A$. So both set are the complements of each other (without the boundary) and their union is the full space. Can I show that the boundary $\partial A=\partial B$ of the two sets 
 have Lebesgue measure zero? Or does it needs more info about the sets?
 A: Here is a counterexample based on the fat cantor set $C$, see http://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set: 
$C$ is obtained interatively from the unit intervall $[0,1]$; In the first step one removes a certain open intervall from the middle of $[0,1]$. In the second step one removes certain open Intervalls from the middle of the two new intervalls, and so on. For the details see the above link. $C$ is defined as the set of limit points of this process and can be written as the infinite intersection of compact sets and is hence compact. Also it has positive measure and $C^c$ is dense in $\mathbb R$ by its construction. 
Now define 
$U_1 = \mathbb R \setminus [0,1] \cup \bigcup_{n \geq 1} \{\text{points removed in the 2nth step}\}$ and $U_2 = \bigcup_{n \geq 1}\{\text{points removed in the (2n - 1)th step}\}$.
$U_1$ and $U_2$ are clearly open, disjoint and $U_1 \cup U_2 = C^c$. Also for each point $a \in C$ for all $i \geq 1$ there exists precisely one closed intervall $I_i$ obtained in the $i$th step of the construction. It follows $a = \bigcap_{i \geq 1} I_i$. Since each $I_i$ contains points of $U_1$ as well as $U_2$ it follows that $C = \partial U_1 = \partial U_2$.
Setting $A = \overline U_1$ and $B = \overline U_2$ we have $A \cup B = \mathbb R$, $\partial A = \partial B$, $A^c = B\setminus \partial B$, $B^c = A \setminus \partial A$ and $\partial A = \partial B$ has positive measure.
