For any subset $A$ of a given metric space, there is a specific dimension $\alpha$ (the Hausdorff dimension of $A$) for which the $\beta$-dimensional Hausdorff measure of $A$ is $\mathcal{H}^{\beta}(A)=+\infty$ for $\beta < \alpha$ and it is $\mathcal{H}^{\beta}(A)=0$ for $\beta > \alpha$.