We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. First we define a real motivic integral which admits a change of variable formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for generically arc-analytic maps. Then we characterize in terms of the motivic measure, germs of arc-analytic homeomorphisms between real algebraic varieties which are bi-Lipschitz for the inner metric. (Based on a joint paper with J.-B. Campesato, T. Fukui, and K. Kurdyka.)