Integration in algebraically closed valued fields

The first two steps of the construction of motivic integration in the fundamental work of Hrushovski and Kazhdan (2006) [8], have been presented in Yin (2010) [12], . In this paper we present the final third step. As in Yin (2010) [12], we limit our attention to the theory of algebraically closed valued fields of pure characteristic 0 expanded by a -generated substructure S in the language LRV. A canonical description of the kernel of the homomorphism L is obtained.