Denote $\mathcal L^0$ as the set of all rvs (i.e. including simple rvs etc..) $\mathcal L^0:=\mathcal L^0(\Omega,\mathcal{F})$, where $\mathcal L^0$ is a vector space. We define a rv in $\mathcal L^0$ as $$ E(X): = \sup\{\hat{E}(Y):Y\le X \}$$ where $Y$ are contained in the set of all simple random variables. Hence this set is monotonic […]

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## Expectations in vector space $L^0$

- Post author By Q+A Expert
- Post date March 31, 2020
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- Tags \forall X\leq Y$ $E(aX+bY) = aE(X) + bE(Y)$ For the first one i think we should prove $\sup\{\hat{E}(Y):Y\le X \} \le \sup\{\hat{E}(X):X\le, \mathcal{F}, a, b0, Denote $\mathcal L^0$ as the set of all rvs (i.e. including simple rvs etc..) $\mathcal L^0:=\mathcal L^0(\Omega, that $E(X)\leq E(Y), where $\mathcal L^0$ is a vector space. We define a rv in $\mathcal L^0$ as $$ E(X): = \sup\{\hat{E}(Y):Y\le X \}$$ where $Y$ are contained, Y \in \mathcal L^0