This paper seeks to develop an integration theory based on the classical Henstock–Kurzweil integral ([4]) for abstract measure spaces. More specifically, the objective is to define an HK-type integral in measure spaces that admit a fractal structure, which are characterized by their recursive behavior. These structures were introduced by M.A. Sánchez-Granero and further explored by J.F. Gálvez-Rodríguez to construct measures within such frameworks ([1], [2]).
First, we define the classes of partitions of the space, which are formed by the elements of the fractal structure, and set up the concept of gauge as a function γ : X → N. In order to guarantee the existence of these partitions, an analogous result to Cousin’s Lemma for compact intervals of Rm is proved. Furthermore, the relationship between the classical theory and this fractal integral in Rm is demonstrated by utilizing a natural fractal structure
defined on compact intervals.
Secondly, we recover fundamental properties of the classical HK-integral, such as Cauchy’s Criterion and Henstock’s Lemma, by assuming additional conditions on the underlying fractal structure. Additionally, we prove that the Monotone and Dominated Convergence Theorems hold for this fractal HK-type integral.
Finally, an application of this integral in Linearly Ordered Topological Spaces (LOTS) ([3]) is provided. We present the construction of a fractal structure in a second-countable LOTS that satisfies desirable properties for this integration theory, leading to a version of the Fundamental Theorem of Calculus for LOTS.
