In this paper, we construct a theory of integration of Voevodsky motives over a perfect field $k$, and show that it circumvents some of the complications of motivic integration, leading to new arithmetic and geometric results concerning K-equivalent $k$-varieties. One main application is that up to direct summing a common Chow motive, K-equivalent smooth projective $k$-varieties have the same $\mathbb{Z}[1/p]$-Chow motives ($p$ is the characteristic exponent of $k$), partially answering a conjecture of Chin-Lung Wang. In addition to generalizing a theorem of Kontsevich on the equality of Hodge numbers of K-equivalent smooth projective complex varieties, we show that such varieties have isomorphic \textit{integral} singular cohomology groups. On the arithmetic side, we show that K-equivalent smooth $k$-varieties have isomorphic $\ell$-adic Galois representations up to semi-simplification. Furthermore, we connect this theory of integration of Voevodsky motives to the existence of motivic 查看全文>>