During my research, I came along a nice lemma which looks like a Birkhoff-Neumann-theorem result, but in a version for stochastic matrices. Namely, I have:
Lemma. Let $M$ be a stochastic matrix, then $M$ can be written as a convex combination of stochastic matrices with entries in $\{0,1\}$.
This looks similar to the Birkhoff-Neumann-theorem and the proof is not difficult either, so I have the feeling, that this result has occurred somewhere before. Unfortunately, I did not find any concrete references for it.
Maybe it's just folklore, but can someone point me, where this result may already have appeared?