  In \cite{HK}, an integration theory for valued fields was developed with a
Grothendieck group approach. It was shown that the semiring of semi-algebraic
sets with measure preserving morphisms is isomorphic to a certain semiring
formed out of twisted varieties over the residue field and rational polytopes
over the value group. With a view to representation-theoretic applications, we
require a simpler description of the possible values of the integration, and in
particular natural homomorphisms into fields. In the present paper we obtain
such results after tensoring with Q. Since this operation trivializes the full
semiring, we restrict to bounded sets. We show that the resulting Q-algebra is
generated by its one-dimensional part. In the "geometric" case, i.e. working
over an elementary submodel as a base, we determine the structure precisely. As
a corollary we obtain useful canonical homomorphisms in the general case.
  In the appendix we define the Iwahori Hecke algebra of $SL_2$ over an
algebraically closed valued field using motivic integration (to replace the
Haar measure) and study its structure.
