Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero

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.