Inclusion Properties of Henstock-Orlicz Spaces

Abstract

Henstock-Orlicz spaces were generally introduced by Hazarika and Kalita in 2021. In general, a function is Lebesgue integral if only if that function and its modulus are Henstock-Kurzweil integrable functions. Moreover, suppose a function is a finite measurable function with compact supports. In that case, the function is a Henstock-Kurzweil integrable function if only if the function is a Lebesgue integrable function. Due to these properties, Henstock-Orlicz spaces were constructed by utilizing Young functions. This definition is almost similar to the definition of Orlicz spaces, but by embedding the Henstock-Kurzweil integral, and the norm used is the Luxembourg norm. Therefore, an analysis of properties in these spaces is needed carried out more deeply. This research was using a literature study on inclusion properties from scientific journals, especially those related to the Orlicz Spaces. And based on the definition of Henstock-Orlicz spaces and its norm, we formulate a hypothesis regarding the inlcusion properties. By deductive proof, we proof the hypothesis and state it as theorem. In this study, we obtain sufficient and necessary conditions for the inclusion properties in Henstock-Orlicz spaces.