Definition of fractal measures arising from fractional calculus

The sets and curves of fractional dimension have been constructed and found to be useful at number of places in science [1]. They are used to model various irregular phenomena. It is wellknown that the usual calculus is inadequate to handle such structures and processes. Therefore a new calculus should be developed which incorporates fractals naturally. Fractional calculus, which is a branch of mathematics dealing with derivatives and integrals of fractional order, is one such candidate. The relation between ordinary calculus and measures on IR is wellknown. For example, an n-fold integration gives an n-dimensional volume. Also, the solution of df/dx = 1[0,x], where 1[0,x] is an indicator function of [0, x], gives length of the interval [0, x] [2]. The aim of this paper is to arrive at a definition of a fractal measure using the concepts from the fractional calculus. Here we shall restrict ourselves to simple subsets of [0, 1] and more rigorous treatment will be given elsewhere. We first define a differential of fractional order α (0 ≤ α ≤ 1) as follows: dx = d1dx(x)/dx −α where df(x) [d(x− a)]q = 1 Γ(−q) ∫ x