The theory of graph limits is only understood to a somewhat satisfactory 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 one of 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 (generalizing the finite node set and the counting measure on the edge set). Our results may be considered as extensions of flow theory for directed graphs to the measurable case.

只有在密集图和有界图的情况下，才可以在一定程度上理解图极限的理论。但是，在中间案例中有很多兴趣。在一般情况下，图极限的最重要组成部分似乎将是马尔可夫空间（具有固定分布的可测量空间上的马尔可夫链）。这激发了我们的目标，即将一些重要的定理从有限图扩展到马尔可夫空间，或更广泛地扩展到可测空间。在本文中，我们证明了图论中最重要的领域之一-流动理论的大部分可以扩展到可测空间。令人惊讶的是，甚至不完全需要马尔可夫空间结构来获得以下结果：我们只需要一个标准Borel空间，该空间在其平方上有一个度量（一般化有限节点集和计数集在边集上）。我们的结果可能被认为是有向图流理论在可测情况下的扩展。