Conditional Reverse Fatou's Lemma

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

Let $X$ be an integrable random variable.

Let $\sequence {X_n}_{n \in \N}$ be an sequence of non-negative integrable random variables such that:


 * there exists an integrable random variable $Y$ such that:


 * $\size {X_n} \le Y$ almost surely.

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

Let $\ds \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG}$ be a version of the conditional expectation of $\ds \limsup_{n \mathop \to \infty} X_n$ conditioned on $\GG$.

Then:


 * $\ds \limsup_{n \mathop \to \infty} \expect {X_n \mid \GG} \le \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG}$ almost surely.

Proof
We should first verify that a version of the conditional expectation of $\ds \limsup_{n \mathop \to \infty} X_n$ conditioned on $\GG$ exists.

We have:


 * $-Y \le X_n \le Y$

and so:


 * $\ds -Y \le \limsup_{n \mathop \to \infty} X_n \le Y$ almost surely

so that:


 * $\ds -\infty < -\expect Y \le \expect {\limsup_{n \mathop \to \infty} X_n} \le \expect Y < \infty$

from Expectation is Monotone.

So:


 * $\ds \limsup_{n \mathop \to \infty} X_n$ is integrable

and we have that conditional expectation of $\ds \limsup_{n \mathop \to \infty} X_n$ conditioned on $\GG$ exists.

Now, we have that:


 * $Y - X_n \ge 0$

So, we can apply Conditional Fatou's Lemma to obtain:


 * $\ds \expect {\liminf_{n \mathop \to \infty} \paren {Y - X_n} \mid \GG} \le \liminf_{n \mathop \to \infty} \expect {Y - X_n \mid \GG}$ almost surely.

We have, by Conditional Expectation is Linear:


 * $\ds \expect {Y - X_n \mid \GG} = \expect {Y \mid \GG} - \expect {X_n \mid \GG}$

So we have:

Similarly:

So we have:


 * $\ds \expect {Y \mid \GG} - \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG} \le \expect {Y \mid \GG} - \limsup_{n \mathop \to \infty} \expect {X_n \mid \GG}$ almost surely.

So:


 * $\ds \limsup_{n \mathop \to \infty} \expect {X_n \mid \GG} \le \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG}$ almost surely.