Conditional Expectation is Monotone

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ and $Y$ be integrable random variables on $\struct {\Omega, \Sigma, \Pr}$ such that:

$X \le Y$ almost everywhere.

Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra of $\Sigma$.

Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.

Let $\expect {Y \mid \GG}$ be a version of the conditional expectation of $Y$ given $\GG$.

Then we have:

$\expect {X \mid \GG} \le \expect {Y \mid \GG}$ almost everywhere.

Proof

We have:

$Y - X \ge 0$ almost everywhere.

So, for each $A \in \GG$ we have:

$\paren {Y - X} \cdot 1_A \ge 0$ almost everywhere.

So, from Expectation is Monotone:

$\expect {\paren {Y - X} \cdot 1_A} \ge 0$

for each $A \in \GG$.

So, from Condition for Conditional Expectation to be Almost Surely Non-Negative, we have:

$\expect {Y - X \mid \GG} \ge 0$ almost everywhere.

Then, from Conditional Expectation is Linear:

$\expect {Y \mid \GG} - \expect {X \mid \GG} \ge 0$ almost everywhere.

So that:

$\expect {X \mid \GG} \le \expect {Y \mid \GG}$ almost everywhere.

$\blacksquare$