ABSTRACT
An object-free definition of a category is always possible. So, to define measures over any category, an appropriate additive structure should exist over the arrows or morphisms of any arbitrary category. However, every category is not additive, for arrows to satisfy conditions of associations of the definition of additivity may not exist, in general. In particular, a zero arrow may not exist. From the families of the objects of any category, an additive category can always be constructed. Even when the categorical concept of a subobject of an object is analogous to that of the subset of a set, the notion of a subobject is not naturally available within the most general framework of category theory. To provide categorical foundations to measure theory, we need to first identify appropriate categorical aspects of the usual, construction of the measures. The concept of a coproduct in the case of the category of sets and functions is that of the disjoint union.
