  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 type of non-archimedean 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 L_TRV. 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.
