Conditional Monotone Convergence Theorem

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

Let $X$ be an non-negative integrable random variable.

Let $\sequence {X_n}_{n \in \N}$ be an increasing sequence of non-negative integrable random variables converging almost surely to $X$.

For each $n \in \N$, let $\expect {X_n \mid \GG}$ be a version of the conditional expectation of $X_n$ conditioned on $\GG$.

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

Then:


 * $\ds \lim_{n \mathop \to \infty} \expect {X_n \mid \GG} = \expect {X \mid \GG}$ almost everywhere.