I have a statement that reads:
If $Z_1,Z_2$ are random variables such that $Z_1 \geq Z_2$, then $\rho(Z_1) \geq \rho(Z_2)$.
where $\rho$ is a function.
What is the meaning of $Z_1 \geq Z_2$? I am particularly interested in the monotonicity of coherent risk measures. For discrete random variables, say:
$\Pr[Z_1=x_i] = p_i$ and $\Pr[Z_2=y_j] = q_j$
how can I write the condition $Z_1 \geq Z_2$ using $x_i,p_i,y_j,q_j$?