  This article explores some properties of universal covers of compact Kahler
manifolds, under the assumption of Caratheodory measure hyperbolicity. In
particular, by comparing invariant volume forms, an inequality is established
between the volume of canonical bundle of a compact Kahler manifolds and the
Caratheodory measure of its universal cover (similar result as in [Kikuta 10]).
Using similar method, an inequality is established between the restricted
volume of canonical bundle of a compact Kahler manifolds and the restricted
Caratheodory measure of its covering, solving a conjecture in [Kikuta 13].
