Lifts of Borel actions on quotient spaces

The paper’s Theorem 1.1 states precisely that if E is compressible, then every Borel action G ↷ X/E admits a class-bijective lift G ↷ (X,E) . The mechanism runs through links: Proposition 3.4 shows that an (E,E∨G)-link exists if and only if there is a class-bijective lift , and Theorem 3.6 proves that whenever E ⊆ F are compressible CBERs there is a smooth (E,F)-link . The candidate’s solution follows exactly this blueprint: it defines F = E∨G, uses compressibility of E to infer compressibility of F, invokes the existence of a smooth (E,F)-link, and then uses the link to define the desired action by selecting the unique point in [x]L ∩ (g·[x]E). This matches the paper’s proof structure and content. One minor slip: the writeup momentarily asserts “if x E x′ then x L x′,” which is not required and generally false under the link definition; nevertheless, the construction of T_g and the class-bijective action remains correct because T_g(x), T_g(x′) lie in the same E-class anyway. Overall, both are correct and essentially the same proof.