Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
Motivic integration has been introduced in the talk of Maxim Kontsevich at Orsay in 1995. This talk also dealt with orbifold cohomology as well as 2 related papers of Lev Borisov?. The orbifold cohomology has been continued by Weimin Chen, Yongbin Ruan and collaborators, and later also by algebraic geometers Abramovich, Vistoli, and others. From physical side a pioneer of both subjects is also Batyrev.
Later, more general framework of motivic integration in model theory has been put forward by Denef and Loeser, partly based on Denef’s work on $p$-adic integration. More recent work using model theoretical approach is by Hrushovski and Kazhdan.
Jan Denef, François Loeser, Definable sets, motives and $p$-adic integrals, J. Amer. Math. Soc. 14 (2001), no. 2, 429–469, doi
Jan Denef, François Loeser, Motivic integration and the Grothendieck group of pseudo-finite fields Proc. ICM, Vol. II (Beijing, 2002), 13–23, Higher Ed. Press, Beijing, 2002.
R. Cluckers, F. Loeser, Constructible motivic functions and motivic integration, Invent. Math. 173 (2008), 23–121 math.AG/0410203
Jan Denef, François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232.
D. Abramovich, M. Mariño, M. Thaddeus, R. Vakil, Enumerative invariants in algebraic geometry and string theory, Lectures from the C.I.M.E. Summer School, Cetraro, June 6–11, 2005. Edited by Kai Behrend and Marco Manetti. LNIM 1947, Springer 2008. x+201 pp.
Manuel Blickle, A short course on geometric motivic integration, math.AG/0507404
Ehud Hrushovski, David Kazhdan, Motivic Poisson summation, arxiv/0902.0845
Ehud Hrushovski, David Kazhdan, The value ring of geometric motivic integration and the Iwahori Hecke algebra of $SL_2$, math.LO/0609115; Integration in valued fields, in Algebraic geometry and number theory, 261–405, Progress. Math. 253, Birkhäuser Boston, pdf
Julia Gordon, Yoav Yaffe, An overview of arithmetic motivic integration, arxiv/0811.2160
Thomas C. Hales, What is motivic measure?, math.LO/0312229
David Kazhdan, Lecture notes in motivic integration, with intro to logic and model theory, pdf
R. Cluckers, J. Nicaise, J. Sebag (Editors), Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry, 2 vols. London Mathematical Society Lecture Note Series 383, 384
Raf Cluckers, Motivic integration for dummies, pdf, A course on motivic integration I, II, III
Raf Cluckers, Julia Gordon, Immanuel Halupczok, Motivic functions, integrability, and uniform in p bounds for orbital integrals, arxiv/1309.0594
Lou van den Dries, Lectures on Motivic Integration , Ms. University of Illinois at Urbana-Champaign. (dvi)
Julien Sebag, Intégration motivique sur les schémas formels, Bull. Soc. Math. France 132 (2004), no. 1, 1–54, MR2005e:14017
Takehiko Yasuda, Motivic Integration over Deligne-Mumford Stacks , arXiv.0312115 (2004). (pdf)
M. Larsen, Valery Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85–95, 259, math.AG/0110255, MR2005a:14026, journal; Rationality criteria for motivic zeta functions, Compos. Math. 140 (2004), no. 6, 1537–1560, math.AG/0212158
Alexei Bondal. M. Larsen, Valery Lunts, Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. 2004, no. 29, 1461–1495, math.AG/0401009
Emmanuel Bultot, Motivic integration and logarithmic geometry, PhD thesis arxiv/1505.05688