2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69477The aim of this article is to develop the theory of motivic integration over Deligne-Mumford stacks and to apply it to the birational geometry of stacks.60 pages ; v.2, minor changes, references added ; v.3, a largely revised version. The base field is now assumed only to be perfect ; v.4, rewritten in detail. the set in which integrals take values is now the semiring of equivalence classes of convergent stacks. a result on the discrepancy of quotient stacks has been added; v.5, very slight modification. some historical accounts have been correctedAlgebraic GeometryPrimary 14A20; Secondary 14B10, 14E15text