单侧盒几何与金字塔不变量到gd-集和qm-空间的扩展
Extensions of One-Sided Box Geometry and Pyramid Invariants to gd-Sets and qm-Spaces
AI总结：
该研究将单侧盒几何与金字塔不变量扩展到gd-集和qm-空间，探讨了相关收敛性质、可观测量恢复及测度关系，完善了该领域的几何与概率分析。
AI中文摘要：
gd-集的支配关系（记录哪个函数族近似哪个）在盒收敛下是封闭的。更一般地，在渐近满测度子集上的近似1-利普希茨映射，其限制测度在推向后收敛到目标测度，会将目标表示置于源金字塔的每个子序列弱极限中。对于弱收敛的金字塔，可观测量的有限有序元组的联合分布，在其质量参数向右扰动后可恢复可观测量直径，在几个正质量集的质量参数共同向左扰动后可恢复其中最大的公共前向间隙；当扰动消失时，上下限公式一致。在不假设gd-集函数族封闭的情况下，我们比较可观测量直径、前向分离的非负部分以及上下中位数尾质量；当且仅当每个正半径处的中位数尾质量均消失时，所有可观测量在测度上与合适的常数一致接近。
英文摘要：
Domination of gd-sets, the relation recording which function family approximates which, is closed under box convergence. More generally, approximate 1-Lipschitz maps on asymptotically full-measure subsets, whose restricted measures converge to the target measure after pushforward, place the target representation in every subsequential weak limit of the source pyramids. For weakly convergent pyramids, bounded joint distributions of finite ordered tuples of observables recover both observable diameter, after rightward perturbation of its mass parameter, and the largest common forward gap among several positive-mass sets, after common leftward perturbation of their mass parameters. The lower- and upper-limit formulas agree as the perturbations vanish. Without closure assumptions on a gd-set's function family, we compare observable diameter, the nonnegative part of forward separation, and upper and lower median-tail masses. All observables become uniformly close in measure to suitable constants exactly when both median-tail masses vanish at every positive radius.
