Generalized Euler characteristic in power-bounded T-convex valued fields

We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski-Kazhdan style motivic integration for certain types of nonarchimedean o-minimal fields, namely power-bounded T-convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the VF-sort and the RV-sort of the language LTRV. Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits not just one, but two specializations to Z. The overall structure of the construction is modeled on that of the original Hrushovski-Kazhdan construction.