Given two measurable disjoint subsets $A,B\subset\Bbb R^n$, which are surfaces of some dimension $k<n$ (e.g. two curves in $\Bbb R^2$ or in $\Bbb R^3$) of finite $k$-dim Lebesgue measure such that $$ d_H(A,B)<\epsilon $$ where $d_H$ is the Hausdorff distance.
Can we deduce that $$ |\mu_k(A)-\mu_k(B)|<\epsilon $$ ?
On
I'll give my example.
The $0$-dimensional measure is just the usual counting measure. In $\mathbb{R}$, consider $A=\{0\}$ and $B$ a finite set of points in $[-\epsilon,\epsilon]-\{0\}$. Then $d_H(A,B)\leq\epsilon$ but $\#A=1$ and $\#B$ is arbitrary.
Now you can take product with $[0,1]^k\times\{0\}^{n-k-1}$ to get example for any $k<n$.
Take $k = 1$ and $n=2$. Let $A = [0] \times \{i/N \mid i = 0,1,\ldots, N\}$ and $B = ([0] \times [0,1]) \backslash A$. Then $d_H(A,B) \to 0$ as $N \to \infty$, but $A$ has $1$-dimensional Lebesgue measure $0$ and $B$ has $1$.