Geodesic
Some Generalizations of the Notion of Length
Matematičeskie zametki, Tome 70 (2001) no. 6, pp. 875-881

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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. 
@article{MZM_2001_70_6_a5,
     author = {B. A. Kats},
     title = {Some {Generalizations} of the {Notion} of {Length}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {875--881},
     publisher = {mathdoc},
     volume = {70},
     number = {6},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_70_6_a5/}
}
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B. A. Kats. Some Generalizations of the Notion of Length. Matematičeskie zametki, Tome 70 (2001) no. 6, pp. 875-881. http://geodesic.mathdoc.fr/item/MZM_2001_70_6_a5/