Harmonizing Dominance: Exploring the Order McShane and the Order Weak-McShane Integrals in Banach Lattices

It is widely recognized that the McShane integral of real-valued functions, a Riemann-type integral, is equivalent to the Lebesgue integral. Building on this foundational equivalence, McShane's unique approach to Lebesgue's integration theory has since been extended to the domain of vector-valued functions. For functions taking values in a Banach space, the integral of choice is the Bochner integral. This integral is known to be stronger than the Birkhoff integral, and the Birkhoff integral is stronger than the McShane, Pettis, and Dunford integrals. Following this, the development of order-type integrals has emerged, especially for functions valued in ordered vector spaces, and more specifically, in Banach lattices, which accommodate a structured approach to integration based on the order of elements. In this paper, we introduce an alternative definition of the weak McShane integral and explore McShane-type integrals within the realm of functions defined on a compact metric Hausdorff topological space, where these functions take values in a Banach lattice with an order-continuous norm. Our investigation encompasses both the norm and the order structure of the space. Throughout this study, we derive several noteworthy outcomes that illuminate the relationships between these integrals. This is particularly evident in scenarios where the Banach space does not contain any copy of c0, while proving the equivalence of the weak order-type McShane and order Pettis integrals.