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

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. Rogers's book Hausdorff Measures

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 $d>1$.

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