  We construct explicit examples of Frostman-type measures concentrated on
arbitrary planar rectifiable curves of positive length. Based on such
constructions we obtain for each $p \in (1,\infty)$ an exact description of the
trace space of the first-order Sobolev space $W^{1}_{p}(\mathbb{R}^{2})$ to an
arbitrary planar rectifiable curve $\Gamma \subset \mathbb{R}^{2}$ of positive
length.
