  We develop a theory of integration over valued fields of residue
characteristic zero. In particular we obtain new and base-field independent
foundations for integration over local fields of large residue characteristic,
extending results of Denef,Loeser, Cluckers. The method depends on an analysis
of definable sets up to definable bijections. We obtain a precise description
of the Grothendieck semigroup of such sets in terms of related groups over the
residue field and value group. This yields new invariants of all definable
bijections, as well as invariants of measure preserving bijections.
