A <i>p</i>-adic variant of Kontsevich–Zagier integral operation rules and of Hrushovski–Kazhdan style motivic integration

Abstract We prove that if two semi-algebraic subsets of ℚ p n {\mathbb{Q}_{p}^{n}} have the same p -adic measure, then this equality can already be deduced using only some basic integral transformation rules. On one hand, considered as a positive answer to analogue question asked by Kontsevich–Zagier in reals (though is much harder). other our result also stating over {\mathbb{Q}_{p}} , universal motivic integration (in sense Hrushovski–Kazhdan) coincides with usual integration.