Starting from the Golab Theorem, which states that in a metric space \((Q,d)\) the Hausdorff measure \({\mathcal H}^1_d\), when restricted to the class of the compact connected subsets of \(Q\), is lower semicontinuous for the Hausdorff distance between sets, it is shown that actually a more general result holds for the \(\Gamma\)-convergence of sequences of Hausdorff measures \(\{{\mathcal H}^1_{d_j}\}\), each one depending on a proper metric \(d_j\), with respect to the uniform convergence of \(\{d_j\}\) on \(Q\times Q\). In particular, it is proved that the \(\Gamma\)-convergence of \({\mathcal H}^1_{d_j}\) to \({\mathcal H}^1_d\) is equivalent to the uniform convergence on \(Q\times Q\) of \(\{d_j\}\) to the metric \(d\). Some results are obtained in the larger framework of semi-distances, where some interesting facts occur. Examples and open problems are presented as well, mainly dealing with the homogenization of distance functions.