We develop a theory of motivic integration for smooth rigid varieties. As an application we obtain a motivic analogue for rigid varieties of Serre's invariant for $p$-adic varieties. Our construction provides new geometric birational invariants of degenerations of algebraic varieties. For degenerations of Calabi-Yau varieties, our results take a stronger form.
"Motivic integration on smooth rigid varieties and invariants of degenerations." Duke Math. J. 119 (2) 315 - 344, 15 August 2003. https://doi.org/10.1215/S0012-7094-03-11924-9