For $A\subset \mathbb R^N$, if $m(A) = 0$ and $\mathcal H^{N-1}(\partial A) = 0$, is there anything more can be said about $A$?
Here $m$ is the Lebesgue measure on $\mathbb{R}^N$ and $\mathcal H^{N-1}$ is the same as Lebesgue measure on $\mathbb R^{N-1}$.
Thank you very much!
Edit: Here is a bigger picture of the problem. I have done some calculation and the result is in the form of $$\int_A \nabla u \cdot \nabla v \;dm $$ where $A$ is the level set $A:= \{|\nabla u(x) | = \|\nabla u\|_\infty\}$.
And I would like to give meaning to the degenerate case $m(A) = 0$. Using Divergence theorem, we get $$\int_A \nabla u \cdot \nabla v \;dm = \int_{\partial A} v(\nabla u \cdot \nu) d\mathcal{H}^{N-1} - \int_A v \triangle u \;dm $$ where $\nu(x)$ is the normal outward vector at $x\in \partial A$. So what I have left is $$\int_{\partial A} v(\nabla u \cdot \nu) d\mathcal{H}^{N-1} .$$ Again I would like to avoid the degenerate case where $\mathcal{H}^{N-1}(\partial A) = 0$, and that was why I asked the above question.
Thank you very much!