How to show a subadditivity of a worst conditional expectation?

Question

I have a definition of worst conditional expectation: Assume that $E[X-]$ is finite, then $WCE(X)=-\mathrm{inf}\{ E[X|A]:P[A]> \alpha \}$ is the worst conditional expectation at level alpha of X. I have to show that it is subadditive that is that for any $X,Y$ we have that $WCE(X+Y) \leq WCE(X)+WCE(Y)$. I know that $\mathrm{inf} (X+Y) = \mathrm{inf}(X) + \mathrm{inf}(Y)$ so I think that it rather should be $WCE(X+Y)=WCE(X)+WCE(Y).$ But does it imply that $WCE(X+Y) \leq WCE(X)+WCE(Y)$ ?

Answer 1

It is not true that $inf(A+B) =inf(A)+inf(B)$. What is true is $inf(A+B) \geq inf(A)+inf(B)$ for any two sets A and B of real numbers. Hence $inf\{E[(X+Y)|A]:P(A)>\alpha\} \geq inf\{E[X|A]:P(A)>\alpha \}+inf\{E[Y|A]:P(A)>\alpha \}$. When you multiply both sides by $(-1)$, $\geq$ becomes $\leq$ and you get the desired result.