Integration over a fractal curve and the jump problem

A definition of integration, i.e., a generalization of a functional of the form
$$ u(z)\mapsto\int_\Gamma f(z)u(z)dz $$
to the case where $\Gamma$ is a fractal curve on the complex plane and $f(z)$ (integration density) is a function defined on this curve is given. The existence and uniqueness of the integral with given density are examined.