Let X and Y be two independent random variables. If I know for some function f that $E_Y[f(X,Y)]$ exists and $P_X[f(X,Y)>t|Y]<\delta$ holds (for any fixed instance of Y) then does it follow that $P_X[E_Y[f(X,Y)]>t]<\delta$?
Here is an attempt, but I'm not sure if it is correct or not:
Define the event $A="f(X,Y)<t"$. We know that $P_X(A)\ge 1-\delta$. Let event B be $B="E_Y[f(X,Y)]<t"$. Now, $A=>B$. Therefore, $P_X(A) <= P_X(B)$, and since $P_X(A)\ge 1-\delta$ we get that $P_X(B)\ge 1-\delta$, which is the same as $P_X(\neg B)<\delta$.