For nonnegative RVs, one can say that if X>Y, then E(X)>E(Y). Now the proof of this is using a simple RV Z, Z>X s.t. E(Z)$\ge$E(x)-$\epsilon$.
I have two simple questions about the definition and proof:
1) What does it mean for X$<$Y? Does it mean stochastically less or greater? i.e. X1 is stochastically larger in than X2 if $P(X1>t)≥P(X2>t)$, Or does it mean X is strictly smaller than Y for all w's.
2) And is $\epsilon$ all positive $\epsilon$>0 in this proof? then how can we gaurantee to find such a simple Z in this proof? I suppose this is related to nonnegative RV's definition but I do not understand the definiton of nonnegative RVs. E(X)+sup{EZ, simple Z $Ω \to I\!R$, Z$\le$X} so I assume X can be continuous here right? than can anyone explain this definition and give me a simple example? I am confused since again beacuse of what Z$\le$X means.
Thanks in advance.