The main goal of the work is to examine, via the uniformly unbounded Lyapunov functions, the global stability of compact random sets in random dynamical systems defined on Banach space. In addition, the possibility of a transition of the advantage of the global asymptotic stability of compact random sets between topologically conjugate random dynamical systems is proved. We also showed that compact random sets can be contained in an attraction region. The necessary condition for a compact connected random set to have asymptotic stability is given. The stability of affine random dynamical systems is analyzed utilizing the second Lyapunov approach.