Consider a well-defined joint distribution over $\mathcal{X} \times \mathcal{Y}$. Does
$$P(Y=f(x) \mid X=x) \geq P(Y = g(x) \mid X=x) \ \ \forall x\in\mathcal{X}$$
always imply
$$P(Y=f(X) \mid X=x) \geq P(Y = g(X) \mid X=x)$$ ?
Intuitively, the second inequality should naturally hold but the proof seems non-trivial to me. Any thoughts?
The two inequalities are equivalent for any $x$ since
$$P(Y=h(X)\mid X=x)=P(Y=h(x)\mid X=x)$$