Let $X$ and $Y$ be random variables, and let $f(x,y)$ and $g(x,y) = \frac{ \partial f }{ \partial x} (x,y)$ be functions.
Suppose that $\mathbb{E} \left( g(X,Y) \mid X \right) \leq 0$.
Can we say (either in general, or under fairly mild conditions) that $\mathbb{E} \left( f(X,Y) \mid X = x \right)$ is weakly decreasing in $x$?
If $X$ and $Y$ are not independent, the conclusion is not always true.
For example, we can take f(x,y) = y - x, X is uniform on [0,1]. When $X < 0.5$, Y = 0, when $X \geq 0.5$, Y = 10000.
If we suppose $X$ and $Y$ are independent, and we define $E[f(X,Y)|X=x] = E[f(X,Y)|X]_{X=x}$
then we have $E[f(X,Y)|X=x] = E[f(x,Y)]$ since $X$ and $Y$ are independent. Now we are wondering if $E[f(x,Y)]$ is weakly decreasing, which is true in some conditions, such as $g$ is bounded