Abstract
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 \(f \mapsto \int_\gamma {f{\text{ }}dz{\text{ }}}\) in the space \(C{\text{(}}\gamma {\text{)}}\); 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.
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