TITLE: Domain whose boundry has non zero volume.
QUESTION [5 upvotes]: Can There be a domain in $\mathbb{R^n}$, for any $n$ such that some domain has non zero boundry volume? I.E. volume of boundry is non zero?
Motivation: 
In some theorems, it is specified that volume of boundary is non zero. But I cannot think of domains where volume of boundry is non zero.
EDIT 
If domain by definition is expected to be open subset of $\mathbb{R^n}$, then I would be looking for such open subsets.
Thank You.

REPLY [6 votes]: Let $(0,1) \cap \mathbb Q = \{ q_1,q_2, \dots\}$. We define $A = (0,1) \cap\bigcup_k (q_k-4^{-k}, q_k+4^{-k})$. By construction we have $\bar A = [0,1]$ but an easy estimate shows that $\lambda( A) < 1$ where $\lambda$ is the Lebesgue measure.
If you want a connected example, you can take $\left(A \times (0,1)\right) \cup \left((0,1) \times (0, 1/2)\right) \subset \mathbb R^2$. This one is actually contractible.