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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 $h$.

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I don’t know enough about it to write up a real answer, but you might take a look at the last two paragraphs of this. –  Brian M. Scott Sep 28 '11 at 7:55
    
@BrianM.Scott: thanks, that's an interesting approach to measure the paths of BM. –  Ilya Sep 28 '11 at 8:09
    
@BrianM.Scott: I will be grateful to you if you will put your comment as an answer –  Ilya Oct 8 '11 at 18:36
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I don’t know enough about Hausdorff measure to write up a real answer, but the last two paragraphs of this look as if they might be useful.

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Felix Hausdorff, in his original 1914 paper, constructs a Cantor set with positive,finite measure for a large class of such functions.

[English translation in Classics on Fractals ]

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By consider easy such functions $h$, we can come up with some easy examples. Take as example the test function $h(r) = cr^d$ for some constant $c>0$. Then any set with nonzero finite Hausdorff measure of dimension $d$ would have nonzero finite measure also with this testfunction. If we let $h$ depend on not only the radius of the balls in the cover but also on their centers, we can get any measure which is absolute continuous to a Hausdorff measure. In this case we can also get a less trivial example by considering $h(x,r)=r^{d(x)}$ where $x$ is the center of the ball. By imitating the Cantor construction but removing intervals at each step of size dependent on their midpoint, it is possible to construct sets of "nonconstant" Hausdorff dimension. If we let $d(x)$ be the local dimension at each point in the resulting set; $h(x,r)=r^{d(x)}$ should give a nonzero and finite measure to the constructed set when measured with this test function.

Wikipedia gives a less intuitive example here; the test function $h(r) = r^2 \log \frac{1}{t} \log \log \log \frac{1}{t}$ gives almost surely $\sigma$-finite measure to the Brownian path in $\mathbb{R}^n$ for $n>2$.

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