Integration theory for vector valued functions and the Radon--Nikodym Theorem in the non-archimedean context

Abstract

In this paper we define non-archimedean measures and integral operators taking values in a locally convex space. We show the relation between these two concept. We define what we called integral function respect to an integral operator. We give necessary and sufficient condition in order to know when a function is integrable with respect to an integral operator. In the second part, we define a kind of absolutely continuous relation between measures in this context. After that, we formulate a type of Radon--Nikodym Theorem between vector measures and a scalar measures which are absolutely continuous.