  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.
