Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees

The paper proves all five parts (i)–(v) of the Main Result for branch subgroups of A•_λ by constructing, via Theorem 5.3, a continuous family of infinite product measures {μ_ω} with which G is nonsingular and ergodic, then applying Kakutani’s criterion (Corollary 5.4) for equivalence/disjointness, Lemma 6.1+Theorem 6.2 for irreducibility, Theorem 6.4 for unitary equivalence/disjointness, and Theorem 7.1 for pairwise weak equivalence; these are assembled in Theorem 6.5 (and explained already in the introduction) . By contrast, the candidate solution makes incorrect or unsubstantiated claims: it assumes μ_ω ≈ λ (contradicted by Corollary 5.4(iv), which shows μ_ω ⊥ λ) and uses this to deduce nonsingularity; it asserts “measure–contracting” for any product measure on weakly branch actions (contradicted by the paper’s counterexamples to universal ergodicity for Bernoulli/product measures), and it claims hyperfiniteness of the orbit relation via tail subrelation without proof. While the Kakutani-based equivalence/singularity part aligns with the paper, the key steps ensuring nonsingularity/ergodicity and irreducibility in the candidate proof are flawed.