Conditional Monotone Convergence Theorem

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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:

\(\ds \int_A Y \rd \Pr\) \(=\) \(\ds \int_A \paren {\lim_{n \mathop \to \infty} \expect {X_n \mid \GG} } \rd \Pr\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \int_A \expect {X_n \mid \GG} \rd \Pr\) Monotone Convergence Theorem (Measure Theory)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \int_A X_n \rd \Pr\) Definition of Conditional Expectation on Sigma-Algebra
\(\ds \) \(=\) \(\ds \int_A X \rd \Pr\) Monotone Convergence Theorem (Measure Theory)

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.

$\blacksquare$


Sources

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