  In this paper we extend the notion of a continuous bundle random dynamical
system to the setting where the action of $\R$ or $\N$ is replaced by the
action of an infinite countable discrete amenable group.
  Given such a system, and a monotone sub-additive invariant family of random
continuous functions, we introduce the concept of local fiber topological
pressure and establish an associated variational principle, relating it to
measure-theoretic entropy. We also discuss some variants of this variational
principle.
  We introduce both topological and measure-theoretic entropy tuples for
continuous bundle random dynamical systems, and apply our variational
principles to obtain a relationship between these of entropy tuples. Finally,
we give applications of these results to general topological dynamical systems,
recovering and extending many recent results in local entropy theory.
