Some introductory notes by Blickle. See also Craw and perhaps also Hales. There are also notes by Blickle in folder AG/Motives, maybe the same notes.
http://ncatlab.org/nlab/show/motivic+integration
Craw: An introduction to motivic integration. On arXiv, and in folder AG/Various
arXiv:1004.4260 The yoga of schemic Grothendieck rings, a topos-theoretical approach from arXiv Front: math.KT by Hans Schoutens We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring, in which we can formulate a form of integration resembling Kontsevich’s motivic integration via arc schemes. Whereas the original construction was via definability, we have translated in this paper everything into a topos-theoretic framework.
It is possible that motivic integration, or at least the notion of (convergence of) power series over the Grothendieck ring, is related to finiteness conditions on simplicial presheaves/simplicial varieties.
arXiv:0912.4887 A note on motivic integration in mixed characteristic from arXiv Front: math.AG by Johannes Nicaise, Julien Sebag We introduce a quotient of the Grothendieck ring of varieties by identifying classes of universally homeomorphic varieties. We show that the standard realization morphisms factor through this quotient, and we argue that it is the correct value ring for the theory of motivic integration on formal schemes and rigid varieties in mixed characteristic. The present note is an excerpt of a detailed survey paper which will be published in the proceedings of the conference “Motivic integration and its interactions with model theory and non-archimedean geometry” (ICMS, 2008).
arXiv:1006.5475 Invariance of orientation data for ind-constructible Calabi-Yau categories under derived equivalence from arXiv Front: math.CT by Ben Davison We study orientation data, as introduced by Kontsevich and Soibelman in order to define well-behaved integration maps from the motivic Hall algebra of 3-dimensional Calabi-Yau categories to rings of motives. We start with an example that demonstrates the role of orientation data in this story, before working through the technical details. We give an account of orientation data in the case of categories of compactly supported sheaves on noncompact Calabi-Yau three-folds. We finally study how this structure behaves under pullbacks along quasi-equivalences of categories, prove Kontsevich and Soibelman’s conjecture regarding this behaviour, and also some stronger theorems regarding flops and more general tilts.
arXiv:1102.3832 Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero from arXiv Front: math.AG by R. Cluckers, F. Loeser 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.
nLab page on Motivic integration