Say I have a sequence of surfaces, $S_n$ in $\mathbb{R}^k$ with $d$-dimensional Hausdorff measure $m_d(S_n)$. Say $S_n$ converges in the Hausdorff distance sense to a surface $S$ with $d$-dimensional Hausdorff measure $m_d(S)$.
You may assume all surfaces in question are compact. Do we have
$ m_d(S_n)\to m_d(S)$?
This is false even for curves in the plane. Consider a sequence of ellipses converging to a line segment. The length of the limit curve drops by the factor of 2. If you want an example of curves without boundary, consider a similar example in 3d space. Or consider curves which are disjoint unions of pairs of concentric circles converging to a single circle.