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

Properties of Caratheodory measure hyperbolic universal covers of compact Kahler manifolds