Condition for Conditional Expectation to be Almost Surely Non-Negative

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

Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.

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 $\GG$.

Then we have:


 * $\expect {X \mid \GG} \ge 0$ almost everywhere




 * $\expect {X \cdot 1_A} \ge 0$ for each $A \in \GG$

where $1_A$ is the characteristic function of $A$.

Sufficient Condition
Suppose that:


 * $\expect {X 1_A} \ge 0$ for each $A \in \GG$.

Then we have, by the definition of the conditional expectation of $X$ given $\GG$:


 * $\expect {\expect {X \mid \GG} 1_A} \ge 0$ for each $A \in \GG$.

Set:


 * $A = \set {\omega \in \Omega : \map {\paren {\expect {X \mid \GG} } } \omega < 0}$

Since $\expect {X \mid \GG}$ is $\GG$-measurable, we have $A \in \GG$ from Characterization of Measurable Functions.

From Expectation is Monotone, we have:


 * $\expect {\expect {X \mid \GG} 1_A} \le 0$

So, we have:


 * $\expect {\expect {X \mid \GG} 1_A} = 0$

Then, from Measurable Function Zero A.E. iff Absolute Value has Zero Integral, we have:


 * $\expect {X \mid \GG} 1_A = 0$ almost everywhere.

But, for all $\omega \in A$, we have:


 * $\expect {X \mid \GG} 1_A < 0$

Hence we have $\map \Pr A = 0$.

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

Necessary Condition
Suppose that:


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

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


 * $\expect {X \mid \GG} \cdot 1_A \ge 0$ almost everywhere.

So, from Expectation is Monotone:


 * $\expect {\expect {X \mid \GG} \cdot 1_A} \ge 0$ for each $A \in \GG$.

From the definition of the conditional expectation of $X$ given $\GG$, we have:


 * $\expect {\expect {X \mid \GG} \cdot 1_A} = \expect {X \cdot 1_A}$

So:


 * $\expect {X \cdot 1_A} \ge 0$ for each $A \in \GG$.