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Given $h:\mathbb{R}_0^+ \to \mathbb{R}_0^+$ increasing and right continuous, the outer measure $\mathcal{H}^h$ in $\mathbb{R}^d$ that assigns to every $E\subset\mathbb{R}^d$ the measure $$\mathcal{H}^h(E) = \lim_{\varepsilon \to 0^+}{ \inf_{ \{G_i\}}{ \sum_{i}{h(\operatorname{d}(G_i))}} },$$ where $\operatorname{d}(\cdot)$ is the diameter of a set and the infima are taken over open coverings of $E$ with diameter less than $\varepsilon$, is called the Hausdorff outer measure in $\mathbb{R}^d$ given by $h$. We can obtain a measure if we restrict the outer measure to the measurable sets in the sense of Caratheodoy.

Taking $h(x) = x^s$, for $s\geq 0$ we get the $s$-dimensional Hausdorff measure. In this case we simply use the notation $\mathcal{H}^s$ for $s\geq 0$ instead of $\mathcal{H}^h$.To assign a dimension to a set, we say that $s_0$ is the Hausdorff dimension of $E$ if $s_0$ is the unique non-negative real number satisfying $$s_0 = \inf{\{ s\in[0,+\infty) : \mathcal{H}^s(E) = 0\}} = \sup{\{ s\in[0,+\infty) : \mathcal{H}^s(E) = +\infty\}}.$$ We use the notation $\operatorname{dim}_{\mathcal{H}}(E)$ for the Hausdorff dimension of $E$. It can happen that $$\mathcal{H}^{\operatorname{dim}_{\mathcal{H}}(E)}(E) = 0 \;\;\text{or}\;\; \mathcal{H}^{\operatorname{dim}_{\mathcal{H}}(E)}(E) = \infty. \qquad (*)$$

My questions are:

1 - Is there a general method to, given $s \geq 0$, construct a set $E$ with Hausdorff dimension $s$ and null $\mathcal{H}^s$ measure? And with infinite $\mathcal{H}^s$ measure?

Some time ago I was told something like it can be proven that, given $h$, there is a set $E$ with null $\mathcal{H}^h$ measure but such that $\mathcal{H}^g(E) = +\infty$ for every $g$ such that $$\lim_{x\to 0^+}{\frac{h(x)}{g(x)}} = 0,$$ where $h$ and $g$ are increasing and right continuous functions.

This is a way of saying that the phenomenon $(*)$ is essentially unsolvable in the sense that is not possible to refine the family of admissible $h$ functions $\{x\mapsto x^s \}_{s\geq 0}$ to get rid of it.

2 - Is there a reference for this more general result of existence of "pathological" sets?

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    $\begingroup$ Once you have a (WLOG bounded) set with positive Hausdorff measure of a particular dimension, then consider infinitely many translates of it to get something of the same dimension and infinite measure. $\endgroup$ – Nate Eldredge Sep 9 '16 at 18:39
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The first reference is Hausdorff's paper, "Dimension und Äußeres Maß"

He constructs a Cantor set $E \subseteq [0,1]$ with $0 < \mathcal H^h(E) < +\infty$ for any sensible $h$. So if $0 < s < 1$ this lets us construct $s$-dimensional sets with zero or infinite $\mathcal H^s$-measure if we use $$ h(x) = x^s \log x\qquad\text{or}\qquad h(x) = \frac{x^s}{\log x} $$ for example. If we want larger $s$ we should be able to use Cartesian powers of these sets in $\mathbb R^d$ with $d>1$.

For more elaborate discussion, see perhaps C. A. Roger's book Hausdorff Measures

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