可测拟阵：基础与极小-极大定理
Measurable Matroids: Foundations and Min--Max Theorems
AI总结：
本文在标准无原子测度空间建立可测拟阵理论，证明其核心定理并给出图论匹配、分解等应用，拓展了经典拟阵理论的测度类比框架。
AI中文摘要：
我们在标准无原子测度空间上建立了拟阵理论的测度论类比。受次模集函数的商收敛框架启发，我们的目标是在有限拟阵理论的极限侧识别合适的可测对象，并发展其基本结构与优化理论。我们证明可测拟阵可通过独立集、基、秩函数和闭包算子得到等价描述，该类包含归一化有限拟阵、图论的圈拟阵，以及划分、嵌套、格路、横截和匹配拟阵的可测类比。我们建立了截断、伸长、直和、对偶和子式的可测版本，证明了Edmonds拟阵交定理和Edmonds-Fulkerson拟阵并定理的可测版本，以及公共基的可达性定理。交定理保留经典极小-极大形式，最大值一般被上确界取代；应用于划分拟阵时，它给出二分图论中可测匹配的精确Hall缺陷公式，以及连续二分b-匹配的极小-极大定理，适用于容量约束传输和指定截面。我们给出使上确界可达的秩扩张准则，对二分图论该准则特化为Lyons和Nazarov的可测完美匹配定理；对可测并，Edmonds-Fulkerson秩公式仍然有效。作为可测并及其可达性定理的主要应用，我们证明了图论的可测Nash-Williams-Tutte定理，刻画了超有限本质生成森林的近似覆盖与填充，并在强化的秩不等式下得到精确分解。
英文摘要：
We develop a measure-theoretic analogue of matroid theory on standard atomless measure spaces. Motivated by the quotient-convergence framework for submodular set functions, our aim is to identify suitable measurable objects on the limit side of finite matroid theory and to develop their basic structural and optimization theory. We prove that measurable matroids admit equivalent descriptions by independent sets, bases, rank functions, and closure operators. The class includes normalized finite matroids, cycle matroids of graphings, and measurable analogues of partition, nested, lattice path, transversal, and matching matroids. We establish measurable analogues of truncation, elongation, direct sum, duality, and minors. We prove measurable versions of Edmonds' matroid intersection theorem and the Edmonds--Fulkerson matroid union theorem, together with an attainment theorem for common bases. The intersection theorem retains the classical min--max form, with the maximum replaced in general by a supremum. Applied to partition matroids, it gives an exact Hall-deficiency formula for measurable matchings in bipartite graphings and a min--max theorem for continuous bipartite $b$-matchings, with applications to capacity-constrained transport and prescribed cross-sections. We give a rank-expansion criterion under which the supremum is attained. For bipartite graphings, this criterion specializes to the measurable perfect matching theorem of Lyons and Nazarov. For measurable union, the Edmonds--Fulkerson rank formula remains valid. As a main application of measurable union and its attainment theorem, we prove measurable Nash-Williams--Tutte theorems for graphings, characterizing approximate coverings and packings by hyperfinite essential spanning forests and obtaining exact decompositions under strengthened rank inequalities.
