Some Generalizations of the Notion of Length

Two numerical characteristics of a nonrectifiable arc $\gamma\subset\mathbb C$ generalizing the notion of length are introduced. Geometrically, this notion can naturally be generalized as the least upper bound of the sums $\sum\Phi(a_j)$, where $a_j$ are the lengths of segments of a polygonal line inscribed in the curve $\gamma$ and $\Phi$ is a given function. On the other hand, the length of $\gamma$ is the norm of the functional $f\mapsto\int_\gamma fdz$ in the space $C(\gamma)$; its norms in other spaces can be considered as analytical generalizations of length. In this paper, we establish conditions under which the generalized geometric rectifiability of a curve $\gamma$ implies its generalized analytic rectifiability.