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 sequence of non-negative integrable random variables converging almost surely to $X$, such that:


 * $X_n \le X_{n + 1}$ almost everywhere for each $n \in \N$.

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

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.

Proof
Note that for almost all $\omega \in \Omega$, $\sequence {\map {X_n} \omega}_{n \mathop \in \N}$ is an increasing real sequence with $\map {X_n} \omega \to \map X \omega$.

From Monotone Convergence Theorem (Real Analysis), we then have that:


 * $\map {X_n} \omega \le \map X \omega$

for almost all $\omega \in \Omega$.

From Conditional Expectation is Monotone, we then have:


 * $\expect {X_n \mid \GG} \le \expect {X \mid \GG}$

Also:


 * $X_n \le X_{n + 1}$ almost everywhere

by hypothesis, so again applying Conditional Expectation is Monotone we have:


 * $\expect {X_n \mid \GG} \le \expect {X_{n + 1} \mid \GG}$

So for each $\omega \in \Omega$, $\sequence {\map {\paren {\expect {X_n \mid \GG} } } \omega}_{n \mathop \in \N}$ is an increasing real sequence bounded above by $\map {\paren {\expect {X \mid \GG} } } \omega$.

So by Monotone Convergence Theorem (Real Analysis):


 * $\ds \lim_{n \mathop \to \infty} \map {\paren {\expect {X_n \mid \GG} } } \omega$ exists for all $\omega \in \Omega$.

So define:


 * $\ds Y = \lim_{n \mathop \to \infty} \expect {X_n \mid \GG}$

Then from Pointwise Limit of Measurable Functions is Measurable, $Y$ is a real-valued random variable.

We want to show that $Y$ is a version of the conditional expectation of $X$ conditioned on $\GG$.

We just need to check that:


 * $\ds \int_A Y \rd \Pr = \int_A X \rd \Pr$

for each $A \in \GG$.

We have:

and hence $Y$ is a version of the conditional expectation of $X$ conditioned on $\GG$ and hence we have:


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