Hausdorff dimension; compact plane set; Hausdorff measure
A compact set $T\subset \bold R^2$ is constructed such that each horizontal or vertical line intersects $T$ in at most one point while the $\alpha$-dimensional measure of $T$ is infinite for every $\alpha \in (0,2)$.
 S. Saks: Theory of the integral
. Dover Publications, 1964. MR 0167578