  We develop a theory of Hrushovski-Kazhdan style motivic integration for
certain type of non-archimedean o-minimal fields, namely polynomial-bounded
T-convex valued fields. The structure of valued fields is expressed through a
two-sorted first-order language L_TRV. We establish canonical homomorphisms
between the Grothendieck semirings of various categories of definable sets that
are associated with the VF-sort and the RV-sort of L_TRV. The groupifications
of some of these homomorphisms may be described explicitly and are understood
as generalized Euler characteristics. In the end, following the
Hrushovski-Loeser method, we construct topological zeta functions associated
with (germs of) definable continuous functions in an arbitrary
polynomial-bounded o-minimal field and show that they are rational. The overall
construction is closely modeled on that of the original Hrushovski-Kazhdan
construction, as reproduced in the series of papers by the present author.
