AI 中文总结
该研究针对ZFC模型上随机泛型的等价关系，结合Popa超刚性等工具证明其正测度限制不可弱归约到本质自由等价关系。
AI 中文摘要
设M为ZFC公理系统中足够大有限片段的可数传递模型，E^B_M是M上随机泛型的等价关系。Smythe证明了E^B_M不可Borel归约到M中任意可数群自由作用的轨道关系，并询问能否移除对目标群的限制。我们证明：若A是M上随机实数的正测度Borel集，F是本质自由的可数Borel等价关系，则从E^B_M在A上的限制（记为E^B_M↾A）到F的每个Borel同态，都将A的一个零测余子集映射到单个F等价类。因此，E^B_M的任何正测度限制都不是本质自由的，甚至无法弱Borel归约到本质自由关系。该证明结合了Thomas由Popa上同调超刚性导出的结论、2标记群的完美族，以及使用两个相继随机实数的富比尼论证。
英文摘要
Let $M$ be a countable transitive model of a sufficiently large finite fragment of ZFC, and let $E^{\mathbb B}_{M}$ be equivalence of random generics over $M$. Smythe proved that $E^{\mathbb B}_{M}$ is not Borel-reducible to the orbit relation of a free action of any countable group belonging to $M$, and asked whether the restriction on the target group can be removed. We prove that if $A$ is a positive-measure Borel set of random reals over $M$ and $F$ is an essentially free countable Borel equivalence relation, then every Borel homomorphism from $E^{\mathbb B}_{M}\upharpoonright A$ to $F$ maps a conull subset of $A$ into a single $F$-class. Hence no positive-measure restriction of $E^{\mathbb B}_{M}$ is essentially free, or even weakly Borel-reducible to an essentially free relation. The proof combines Thomas's consequence of Popa cocycle superrigidity with a perfect family of $2$-marked groups and a Fubini argument using two successive random reals.
