Γ Convergence of Hausdorff Measures

We study the dependence of the Hausdorff measure

on the distance

d d

. We show that the uniform convergence of

d_j d j

to

d d

is equivalent to the

\Gamma Γ

convergence of

\mathcal{H}^1_{d_j} H d j 1

to

\mathcal{H}^1_d H d 1

with respect to the Hausdorff convergence on compact connected subsets. We also consider the case when distances are replaced by semi-distances