Derivation and Integration on a Fractal Subset of the Real Line

Ordinary calculus is usually inapplicable to fractal sets. In this chapter, we introduce and describe the various approaches made so far to define the theory of derivation and integration on fractal sets. In particular, we study some Riemann-type integrals (the s-Riemann integral, the sHK integral, the s-first-return integral) defined on a closed fractal subset of the real line with finite and positive s-dimensional Hausdorff measure (s-set) with particular attention to the Fundamental Theorem of Calculus. Moreover, we pay attention to the relation between the s-Riemann integral, the sHK integral, and the Lebesgue integral with respect to the Hausdorff measure ℋs, respectively, and we give a characterization of the primitives of the sHK integral.