Conditional Fatou's Lemma

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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.

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


Then we have:

$\ds \expect {\liminf_{n \mathop \to \infty} X_n \mid \GG} \le \liminf_{n \mathop \to \infty} \expect {X_n \mid \GG}$ almost everywhere.


Proof

From Conditional Expectation of Measurable Random Variable, we have:

$\expect {X_n \mid \GG} = \expect {\expect {X_n \mid \GG} \mid \GG}$

It therefore suffices to show, from Conditional Expectation is Linear:

$\ds \expect {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} - \liminf_{n \mathop \to \infty} X_n \mid \GG} \ge 0$ almost everywhere.

From Condition for Conditional Expectation to be Almost Surely Non-Negative, we can show that:

$\ds \expect {\paren {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} - \liminf_{n \mathop \to \infty} X_n} \cdot 1_A} \ge 0$

for $A \in \GG$.

We indeed have, for each $A \in \GG$:

\(\ds \expect {\paren {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} - \liminf_{n \mathop \to \infty} X_n} \cdot 1_A}\) \(=\) \(\ds \int_A \paren {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} - \liminf_{n \mathop \to \infty} X_n} \rd \Pr\)
\(\ds \) \(=\) \(\ds \int_A \paren {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} } \rd \Pr - \int_A \liminf_{n \mathop \to \infty} X_n \rd \Pr\) Integral of Integrable Function is Additive: Corollary 2
\(\ds \) \(\ge\) \(\ds \int_A \paren {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} } \rd \Pr - \liminf_{n \mathop \to \infty} \int_A X_n \rd \Pr\) Fatou's Lemma for Integrals
\(\ds \) \(=\) \(\ds \int_A \paren {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} } \rd \Pr - \liminf_{n \mathop \to \infty} \int_A \expect {X_n \mid \GG} \rd \Pr\) Definition of Conditional Expectation on Sigma-Algebra
\(\ds \) \(\ge\) \(\ds 0\) Fatou's Lemma for Integrals

$\blacksquare$


Source of Name

This entry was named for Pierre Joseph Louis Fatou.


Sources

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