测度分解的几何性质
Geometric properties of disintegration of measures
AI总结：
本文从最优传输角度研究测度分解，通过传输计划定义分解映射，分析其正则性与绝对连续性，并给出弱连续性条件及分解为绝对连续测度的刚性条件。
摘要：
本文研究测度分解与概率空间几何性质之间的联系。我们证明了一个分解定理，从最优传输问题的角度处理测度分解。我们考察传输计划的分解，这些传输计划用于定义和研究分解映射。利用这些对象，我们研究测度分解的正则性与绝对连续性。特别地，我们给出了使分解映射弱连续的条件，并由此可获得由该映射给出的测度路径。我们证明了测度分解为绝对连续测度的一个刚性条件。
英文摘要：
In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We look at the disintegration of transport plans, which are used to define and study disintegration maps. Using these objects, we study the regularity and absolute continuity of disintegration of measures. In particular, we exhibit conditions for which the disintegration map is weakly continuous and one can obtain a path of measures given by this map. We show a rigidity condition for the disintegration of measures to be given into absolutely continuous measures.
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