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 ( V F , Γ ) -generated substructure S in the language L RV . A canonical description of the kernel of the homomorphism L is obtained.