Condition for Conditional Expectation to be Almost Surely Non-Negative

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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

if and only if:

$\expect {X \cdot \chi_A} \ge 0$ for each $A \in \GG$

where $\chi_A$ is the characteristic function of $A$.


Proof

Sufficient Condition

Suppose that:

$\expect {X \chi_A} \ge 0$ for each $A \in \GG$.

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

$\expect {\expect {X \mid \GG} \chi_A} \ge 0$ 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:

$\expect {\expect {X \mid \GG} \chi_A} \le 0$

So, we have:

$\expect {\expect {X \mid \GG} \chi_A} = 0$

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

$\expect {X \mid \GG} \chi_A = 0$ almost everywhere.

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

$\expect {X \mid \GG} \chi_A < 0$

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

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

$\Box$

Necessary Condition

Suppose that:

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

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

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

So, from Expectation is Monotone:

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

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

$\expect {\expect {X \mid \GG} \cdot \chi_A} = \expect {X \cdot \chi_A}$

So:

$\expect {X \cdot \chi_A} \ge 0$ for each $A \in \GG$.

$\blacksquare$