Theorem: Conditional Reverse Fatou's Lemma

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable random variable.
Let $\sequence {X_n}_{n \mathop \in \N}$ 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:
{{begin-eqn}}
{{eqn	| l = \liminf_{n \mathop \to \infty} \expect {Y - X_n \mid \GG}
		| r = \liminf_{n \mathop \to \infty} \paren {\expect {Y \mid \GG} - \expect {X_n \mid \GG} }
}}
{{eqn	| r = \expect {Y \mid \GG} + \liminf_{n \mathop \to \infty} \paren {-\expect {X_n \mid \GG} }
		| c = Infimum Plus Constant
}}
{{eqn	| r = \expect {Y \mid \GG} - \limsup_{n \mathop \to \infty} \expect {X_n \mid \GG}
		| c = Negative of Supremum is Infimum of Negatives
}}
{{end-eqn}}
Similarly:
{{begin-eqn}}
{{eqn	| l = \expect {\liminf_{n \mathop \to \infty} \paren {Y - X_n} \mid \GG}
		| r = \expect {Y + \liminf_{n \mathop \to \infty} \paren {-X_n} \mid \GG}
		| c = Infimum Plus Constant
}}
{{eqn	| r = \expect {Y - \limsup_{n \mathop \to \infty} X_n \mid \GG}
		| c = Negative of Supremum is Infimum of Negatives
}}
{{eqn	| r = \expect {Y \mid \GG} - \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG}
		| c = Conditional Expectation is Linear
}}
{{end-eqn}}
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.
{{qed}}
{{Namedfor|Pierre Joseph Louis Fatou|cat = Fatou}}
Category:Conditional Expectation
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