This question seems obvious, but not sure how to prove it.
Let $\mathcal{A}$ be a $\sigma$-algebra, and $X$ be a random variable. Suppose $E(X\mid A)\le1$ for any $A\in\mathcal{A}$, can we conclude that $E(X\mid \mathcal{A})\le1$ almost surely?
This question seems obvious, but not sure how to prove it.
Let $\mathcal{A}$ be a $\sigma$-algebra, and $X$ be a random variable. Suppose $E(X\mid A)\le1$ for any $A\in\mathcal{A}$, can we conclude that $E(X\mid \mathcal{A})\le1$ almost surely?
This is a simple question.
For any $A\in\mathcal{A}$, it holds that
$$ \int E(X\mid\mathcal{A})I_A \, dP=E(XI_A)=P(A)E(X\mid A)\le P(A), $$
hence, $E(X\mid\mathcal{A})\le1$ almost surely.. Thanks to Terry for quick guide!