Let $X, Y$ and $Z$ be real-valued integrable random variables. If we assume that for all $r \in \mathbb{R}$ $$\mathbb{E}\left[ X \middle| Z = r\right] > \mathbb{E}\left[ Y \middle|Z =r \right],$$ do we necessarily have $\mathbb{E}\left[ X \right] > \mathbb{E}\left[ Y \right]$?
This would seem natural, and if the condition would be replaced by something that would partition the probability space countably, there would be no problem. However I'm not really sure how to handle the $\sigma$-algebra generated by events $Z=r$.