Conditional Expectation of Non-Negative Random Variable is Non-Negative

Theorem

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

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

Let $X$ be an integrable random variable such that:

$X \ge 0$ almost everywhere.

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

Then:

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

Proof

From Conditional Expectation is Monotone, we have:

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

From Conditional Expectation of Constant, we have:

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

So:

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

$\blacksquare$

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $9.7$: Properties of conditional expectation: a list