论文标题
在可衡量的空间上流动
Flows on measurable spaces
论文作者
论文摘要
在密集的图和有界度图的情况下，仅在任何非平凡程度上才能将图形限制理论理解。但是，对中间情况有很多兴趣。似乎一般情况下最重要的图形限制成分是马尔可夫空间（马尔可夫空间（Markov链的可测量空间上的马尔可夫链具有固定分布）。 这促使我们的目标是将一些重要的定理从有限图扩展到马尔可夫空间，或者更一般地扩展到可衡量的空间。在本文中，我们表明，图理论中最重要的领域之一可以扩展到可衡量的空间。令人惊讶的是，即使是马尔可夫空间结构也不需要完全以获得这些结果：我们需要一个标准的Borel空间，并在其正方形上进行测量。我们的结果可能被认为是针对可测量情况的定向图的流程理论的扩展。
The theory of graph limits is only understood to any nontrivial degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution). This motivates our goal to extend some important theorems from finite graphs to Markov spaces or, more generally, to measurable spaces. In this paper, we show that much of flow theory, one of the most important areas in graph theory, can be extended to measurable spaces. Surprisingly, even the Markov space structure is not fully needed to get these results: all we need a standard Borel space with a measure on its square. Our results may be considered as extensions of flow theory for directed graphs to the measurable case.
