We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov–Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that $E_1$ is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov–Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided.

我们将解析/Borel 等价关系、轨道等价关系和它们之间的 Borel 约简的概念推广到它们之间的连续和定量对应物：解析/Borel 伪计量学、轨道伪计量学和它们之间的 Borel 约简。我们通过示例激发这些概念，并设置了一些基本的一般理论。我们通过证明 Gromov–Hausdorff 距离保持相同的复杂性来说明新的约简概念，如果它被定义在所有波兰度量空间的类上，空间从下面、从上面、从下面和上面界定。然后我们证明${乙}_{\mathrm{1个}}$不能还原为由轨道伪测量引起的等价，概括了 Kechris 和 Louveau 的开创性结果。我们对 Ben Yaacov、Doucha、Nies 和 Tsankov 提出的关于 Gromov–Hausdorff 和 Kadets 距离中的球是否为 Borel 的问题作出否定回答。在附录中，我们提供了使用博弈的新方法，表明某些伪计量学中的零距离类是 Borel，扩展了 Ben Yaacov、Doucha、Nies 和 Tsankov 的结果。作者有一篇补充论文，其中提供了来自泛函分析和度量几何的最常见伪度量之间的缩减。