In a previous research, we generalized the integration by using the given measure space (ℵ,Γ,M), and θ: ℵ→┴( ) W be a measurable function, where W is an ordered Banach algebra [15].In this research, we have covered one of the fundamental applications for the integration theory which is Radon-Nikodym theorem which by itself is considered essential in the theory of modern probability and other parts of analysis. Several cases for this result have been considered before proceeding into the general theory and the first step in the development of the general Radon-Nikodym theorem is Jordan Hahn decomposition theorem.In this paper, we consider the above W a space with identity element e.
