CONDITIONAL ENTROPY FOR AMENABLE GROUP ACTIONS

The paper states and proves the fiberwise decomposition formulas h^A_µ(T,α) = ∫ h^{A_B}_{µ_B}(T_B,α_B) dµ_β and h^A_µ(T) = ∫ h^{A_B}_{µ_B}(T_B) dµ_β for amenable group actions with an invariant partition β. However, in Theorem 4.7 the proof interchanges a limit with integration without justification; dominated convergence (or an equivalent uniform-integrability argument) is needed at that step, but is not supplied in the text . The model’s solution supplies the dominated-convergence step and a clear derivation of the equality for fixed α, but its “gluing” step for the reverse inequality (Step 5) does not explicitly ensure the resulting global partition α has finite conditional entropy (i.e., lies in Z(C)); this needs a uniform complexity cap or a two-parameter approximation argument. Thus both arguments are essentially correct in idea but incomplete in detail. The main statements align with the paper’s Theorems 4.7 and 4.8, and the existence of entropy limits is grounded in Theorem 4.3 and the Ornstein–Weiss framework , but both proofs as written require minor but nontrivial repairs.