Suppose the conditional distribution of $\theta$ and $x$ satisfies the monotone likelihood ratio property: for every real numbers $\bar{\theta} \ge \theta$ and $\bar{x} \ge x$, we have $ f(\bar{\theta}|\bar{x})/f(\theta|\bar{x}) \ge f(\bar{\theta}|x)/f(\theta|x).$
Now suppose I extend this property as follows: for each real numbers $\bar{x} \ge x$ and each given $\bar{A}$ and $A$ that are subsets of [0,1] with the property that $E[\theta|\bar{x}, \theta \in \bar{A}] \ge E[\theta|\bar{x}, \theta \in A]$, we have $ Pr(\bar{A}|\bar{x})/Pr(A|\bar{x}) \ge Pr(\bar{A}|x)/Pr(A|x).$
If $A$ and $\bar{A}$ are singleton, it is exactly the monotone likelihood ratio property. But I now want to require this property for sets $A$.
What additional condition on the joint distribution of $x$ and $\theta$ will provide this property?