ON INTEGRAL FORMULAS OF METRIC MEAN DIMENSION FOR RANDOM DYNAMICAL SYSTEMS

The paper proves the max–min integral formula for upper metric mean dimension in random dynamical systems (Theorem 1.1) via a convex-analytic route (defining F(μ), separation of convex sets, Stone vector lattices, Daniell–Stone integration, and an averaging-to-invariance step), and its argument is sound under the stated hypotheses, notably the ergodicity of the base which is used to secure the marginal P for the constructed measure. The candidate solution proves the same formula by a direct combinatorial method using maximal separated sets, a log-sum/Jensen inequality, and a measurable selection of maximizers. The approach is standard and essentially correct, though it omits some technical details (notably: a measurable selection of ω↦F^{max}_{ω,n,ε}, a diagonal choice of n_k ensuring simultaneous control of the −f terms, and a precise limsup/liminf handling when passing from (★) to the ε→0 limit). These are fixable with routine tools (e.g., Jankov–von Neumann selection and a two-parameter diagonal argument). Hence both are correct, with substantially different proofs.