A \(p\)-adic variant of Kontsevich-Zagier integral operation rules and of Hrushovski-Kazhdan style motivic integration (Q2238195)

The authors prove that if two semi-algebraic subsets of \(\mathbb Q^n_p\) have the same \(p\)-adic measure, then this equality can already be deduced using only some basic integral transformation rules. On the one hand, this can be considered as a positive answer to a \(p\)-adic analogue of a question asked by Kontsevich-Zagier for the reals [\textit{B. Engquist} (ed.) and \textit{W. Schmid} (ed.), Mathematics unlimited---2001 and beyond. Berlin: Springer (2001; Zbl 0955.00011)]. On the other hand, the result can also be considered as stating that over \(\mathbb Q_p\), the universal motivic integration (in the sense of [\textit{E. Hrushovski} and \textit{D. Kazhdan}, Prog. Math. 253, 261--405 (2006; Zbl 1136.03025)]) coincides with the usual \(p\)-adic integration.