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.