Definition of fractal measures arising from fractional calculus
  It is wellknown that the ordinary calculus is inadequate to handle fractal structures and processes and another suitable calculus needs to be developed for this purpose. Recently it was realized that fractional calculus with suitable constructions does offer such a possibility. This makes it necessary to have a definition of fractal measures based on the fractional calculus so that the fractals can be naturally incorporated in the calculus. With this motivation a definition of fractal measure has been systematically developed using the concepts of fractional calculus. It has been demonstrated that such a definition naturally arises in the solution of an equation describing diffusion in fractal time. 
Kiran M. Kolwankar and Anil D. Gangal