UNIFORM BOREL AMENABILITY IS EQUIVALENT TO RANDOMIZED HYPERFINITENESS

The paper’s Theorem 1 asserts that a bounded-degree Borel graph is uniformly Borel amenable iff it is randomized hyperfinite, and proves it by first building random packings via a strong Connes–Feldman–Weiss-type result (Theorem 6), then converting these packings into random finite equivalence relations; conversely, taking expectations of the uniform measures on random classes gives the uniform amenability kernels (see Definition 1.1 and 1.4, and the proof of Theorem 1 immediately after Theorem 6) . The candidate solution’s “randomized hyperfinite ⇒ uniformly Borel amenable” direction matches the paper’s argument almost verbatim by defining p_n(x,·) as the expectation of the uniform distribution on En(x) . For the difficult direction, “uniformly Borel amenable ⇒ randomized hyperfinite,” the candidate gives a direct, simple quantile-coupling construction (via a Borel selector and a single shared random seed per stage) instead of the paper’s route through tight multipackings and generalized quasi-tilings before invoking Theorem 6; the construction satisfies the same diameter and edge-separation bounds. The minor definability requirements (Borel linear order, Borel selector for finite classes) are standard (e.g., Kechris 1995), so the model’s proof is sound. Hence both are correct, with different proofs for one direction.