  We extend the formalism and results on motivic integration from
["Constructible motivic functions and motivic integration", Invent. Math.,
Volume 173, (2008) 23-121] to mixed characteristic discretely valued Henselian
fields with bounded ramification. We also generalize the equicharacteristic
zero case of loc. cit. by giving, in all residue characteristics, an axiomatic
approach (instead of only using Denef-Pas languages) and by using richer
angular component maps. In this setting we prove a general change of variables
formula and a general Fubini Theorem. Our set-up can be specialized to
previously known versions of motivic integration by e.g. the second author and
J. Sebag and to classical p-adic integrals.
