THE ENTROPY OF AN EXTENDED MAP FOR ABELIAN GROUP ACTIONS

The paper proves h_μ(G, Y | Π_X) = 0 for the natural extension by approximating arbitrary partitions using sets of the form Π_{g,X}^{-1}(B) and a standard small-symmetric-difference-to-small-conditional-entropy lemma, yielding zero relative entropy along Følner sets . The candidate solution instead observes that for the constructed natural extension one has Π_{g,X}^{-1}(A) = Π_X^{-1}(α_g(A)) and hence the σ-algebra on Y is the μ-completion of Π_X^{-1}B_X; therefore every partition is (mod μ) measurable with respect to the conditioning σ-algebra, making all conditional entropies vanish immediately. Both arguments yield the same conclusion. Minor issues in the candidate (a t=g vs. g=e substitution) do not affect correctness. The paper’s proof is sound though longer than necessary; it implicitly contains the key identification Π_{g,X}^{-1}(B) ⊆ Π_{e,X}^{-1}(B) for all g, which already forces the generated σ-algebra to be Π_X^{-1}B_X (hence conditional entropy zero) .