Switching integration and minimization for positive random variables?

Question

Suppose we have three non-negative random variables $X,Y,Z$, they may be dependent and they satisfy $Z\geq\min\{X,Y\}$. Now we know that there exists a constant $A$ such that $E(X)\geq A, E(Y)\geq A$. Is it true that $E(Z)\geq A$?

Answer 1

No. Let $X$ be $1$ with probability $1/2$ and $0$ otherwise (i.e., a fair coin), $Y = 1-X$, and $Z = \min(X,Y) = 0$. Then $E(X) = E(Y) = 1/2$, but $E(Z) = 0$.

Answer 2

Counter example:

Let Z = min(X,Y), with sample (x, y) = (0,0) with probability 1/3, (0,3) with probability 1/3, and (3, 0) with probability 1/3.

E[X] = E[Y] = 1. Z = 0, and E[Z] = 0. Any A between 0 and 1 satisfies the first, but not the last.