Spatial models for boolean actions in the infinite measure-preserving setup

The paper’s Theorem A states that any measure-preserving action of a locally compact Polish group on a standard σ-finite space is spatially isomorphic to a continuous action on a locally compact Polish space with a Radon measure, and it supplies two rigorous proofs: one via boolean actions plus Mackey (yielding Corollary 4.4) and another direct, measured construction (Theorem 5.4) that embeds the entire space with Radon pushforward measure . The candidate solution incorrectly relies on a Varadarajan-style topological realization and then asserts that σ-finite Borel measures on Polish spaces are automatically Radon—this is false in general and precisely why the compact-model approach fails for infinite measures (the pushforward is not locally finite on a compact space) as the paper itself emphasizes . The paper provides the missing local-finiteness/Radon arguments via convolution and G-continuous functions, while the model does not. Hence, the paper correctly solves the problem; the candidate model’s key Radon step is wrong.