Flows on measurable spaces

Abstract The theory of graph limits is only understood to a somewhat satisfactory degree in the cases dense graphs and bounded graphs. There is, however, lot interest intermediate cases. It appears that one most important constituents general case will be Markov spaces (Markov chains on measurable with stationary distribution). This motivates our goal extend some theorems from finite or, more generally, spaces. In this paper, we show much flow theory, areas can extended Surprisingly, even space structure not fully needed get these results: all need standard Borel measure its square (generalizing node set counting edge set). Our results may considered as extensions for directed case.