Convex stochastic order and conditional expectation

Question

I am trying to learn something about stochastic ordering, an idea I just picked up while reading an interesting paper. Let two probability laws $\mu$ and $\nu$ satisfy $\int h d\mu \leq \int h d\nu$ for every convex function $h:[0,1]\rightarrow[0,2]$. Then we write $\mu \leq \nu$ and this is a partial order on probability laws. Now it holds that $\mu \leq \nu$ if and only if there are random variables $Z\sim \nu$ and $Y\sim \mu$ such that $E(Z|Y)=Y$. While the if part is almost trivial, the only if part eludes me and I suspect it is far less trivial. I could not find any helpful text on the subject. If you have an idea on how to proceed or a valuable reference, I’d greatly appreciate it. Cheers!

Answer 1

This is indeed not trivial, and has been establish in Theorem 6 in the paper

L. Rüschendorf, Ordering of distributions and rearrangement of functions, Ann. Probab. 9 (1981), no. 2,276–283.