The New Extensions of the Henstock-Kurzweil and the McShane Integrals of Vector-Valued Functions

In this paper, we define the Hake-Henstock-Kurzweil and the Hake-McShane integrals of Banach space valued functions defined on an open and bounded subset G of m-dimensional Euclidean space R-m. These are "natural" extensions of the McShane and the Henstock-Kurzweil integrals from m-dimensional closed non-degenerate intervals to bounded and open subsets of Rm. Our goal is not a generalization for the sake of generalization. Indeed, we will show theorems which reduce the study of our integrals to the study of McShane and Henstock-Kurzweil integrals. As applications, we will present Hake-type theorems for the Henstock-Kurzweil and the McShane integrals in terms of our integrals.