At my analysis course some time ago we were told that there is definition of Hausdorff measure through the test functions which are continuous and non-decreasing $h:(0,\infty)\to(0,\infty)$ and defined for a subset $E$ of a metric space as $ \mathcal H^h(E) = \lim\limits_{\delta\to 0}\left(\inf\limits_{\Xi(\delta)}\sum\limits_{k}h(r_k)\right) $ where $\inf$ is taken w.r.t. to all at most countable covers of $E$ with closed balls of the radius $r_k\leq\delta$.
If one put $h(r) = r^d$ he has a Hausdorff measure which helps to find the Hausdorff dimension. We were also told that there are examples when set has non-trivial measure with $h$ different from the power function, e.g. logarithmic Hausdorff measure with $h(r)=\min\left(1,\frac1{-\log r}\right)$.
But we weren't told about the examples of sets which admit non-trivial ($\neq0,\neq\infty$) such measure. Do you know any? Not necessary for the logarithmic $h$.