In this paper, we introduce the notion of upper metric mean dimension with potential on any subset (not necessarily compact or invariant) via Carathéodory–Pesin structures. We discuss several possible versions of upper measure-theoretic mean dimensions with potential and find conditions to make these notions coincide. In particular, we present a corresponding variational principle and an inverse variational principle.

In this paper, preimage pressure, which is based on the preimage structure of the system, is defined and studied for random transformations. We obtain analogs of many known results of preimage entropy and preimage pressure for deterministic cases in Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113] and Zeng et al [Pre-image pressure and invariant measures. Ergod. Th. & Dynam. Sys.27 (2007), 1037–1052]. In particular, a variational principle is given and some applications of preimage pressure, such as the investigation of the invariant measures and the equilibrium states, are obtained.