Radon-Nikodym Theorem with Banach Algebra Valued Measure

Abstract

^{In a previous research, we generalized the integration by using the given measure}

^{space (ℵ, Γ,ℳ), and θ: ℵ → ???? be a measurable function, where ???? 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 ???? a space with identity element e.}