Independence from finite independence by Monotone Class Lemma

Question

From the appendix of J.-F. Le Gall: Brownian Motion, Martingales, and Stochastic Calculus

Here are a few consequences of the monotone class lemma that are used above.

2. Let $(X_i : i \in I)$ be an arbitrary collection of random variables, and let $G$ be a $\sigma$-field on the same probability space. In order to show that the $\sigma$-fields $\sigma(X_i : i \in I)$ and $G$ are independent, it is enough to verify that $\sigma(X_{i_1}, ..., X_{i_p})$ is independent of $G$, for any choice of the finite set $\{ i_1,...,i_p \} \subset I$. (Observe that the class of all events that depend on a finite number of the variables $(X_i : i \in I)$, is stable under finite intersections and generates $\sigma(X_i : i \in I)$.

3. Let $(X_i : i \in I)$ be an arbitrary collection of random variables, and let $Z$ be a bounded real variable. Let $i_0 \in I$. In order to verify that $E\left[Z | X_i : i \in I \right]$, it is enough to show that $E\left[Z | X_{i_0},...,X_{i_p} \right]$ for any choice of the finite collection $\{ i_1,...,i_p \} \subset I$. (Observe that the class of all events $A$ such that $E\left[ Z 1_A \right] = E\left[ E\left[ Z | X_{i_0}\right] 1_A \right]$ is a monotone class.)

Despite the hints provided, I am struggling to prove that the two consequences above hold from the Monotone Class Lemma:

1. For item 2 following the hint in the brackets, if I could show that the class $$ \{ A \in \mathcal{F} : \exists\{i_1,...,i_p\} \subset I \text{ such that } A \in \sigma(X_{i_1},...,X_{i_p}) \} $$ is a monotone class ($\lambda$-system), I could also claim it is a $\sigma$-field and the independence would follow. However, the condition that the unions of increasing sets from a monotone class are too in that monotone class turns out to be trickier to verify than I thought. Unfortunately, this answer is not of much help either, as it just assumes the result instead of invoking the lemma.

2. For item 3 following the hint in brackets, I am struggling to make any use of the class $$ \{ A \in \mathcal{F} : E\left[ Z 1_A \right] = E\left[ E\left[ Z | X_{i_0}\right] 1_A \right] \} $$ at all. I thought of restricting $A$ to $\sigma(X_i : i \in I)$ but haven't made much progress. Clearly we are just verifying the definition of conditional expectation in this particular case and even the property I am struggling with in item 2 could probably be proved by the monotone convergence theorem.

Throughout the first problem, I feel like I am not using the proper formulation of the definitions, as none of my attempts work out. I think that once I would have sorted that one out, I could make progress with the second one.

Does anybody have any idea or hint on how to approach the two consequences?

Answer 1

A proof compiled by mimicking the proof of the Uniqueness Lemma for measures on $\pi$-systems (Uniqueness Lemma 1.6 in David Williams' Probability with Martingales, Appendix A1.2).

Define $$ \newcommand{\defeq}{:=} \newcommand{\eqdef}{=:} \mathcal{D} \defeq \{ A \in \mathcal{F} : \forall B \in \mathcal{G} \ \mathbb{P}(A \cap B) = \mathbb{P}(A)\mathbb{P}(B) \}. $$

$$ \pi \defeq \{ A \in \mathcal{F} : \exists \{ i_1, ..., i_p \} \subset \mathcal{I} \text{ such that } A \in \sigma(X_{i_1}, ..., X_{i_p}) \}. $$ Observe that $\pi \subseteq \mathcal{D}$ (as any finite number of $X_i$s is independent from $\mathcal{G}$) and that it is a $\pi$-system. Moreover:

1. $\Omega \in \mathcal{D}$, trivially.

2. For any $B \in \mathcal{G}$ and $A_1, A_2 \in \mathcal{D}$ such that $S_2 \subseteq S_1$ \begin{align*} \mathbb{P}((A_1 \backslash A_2) \cap B) & = \mathbb{P}((A_1 \cap B) \backslash (A_2 \cap B)) \\ &= \mathbb{P}(A_1 \cap B) - \mathbb{P}(A_2 \cap B) \\ &= \mathbb{P}(A_1)\mathbb{P}(B) - \mathbb{P}(A_2)\mathbb{P}(B) \text{ as $A_1, A_2 \in \mathcal{D}$} \\ &= \mathbb{P}(A_1 \backslash A_2)\mathbb{P}(B), \end{align*} so $A_1 \backslash A_1 \in \mathcal{D}$.

3. For any $B \in \mathcal{G}$ and $(F_n)$ an increasing sequence of events from $\mathcal{D}$ such that $F_n \uparrow F$ \begin{align*} \mathbb{P}(F \cap B) &= \mathbb{P}((\lim F_n) \cap B) \\ &= \mathbb{P}(\lim (F_n \cap B)) \\ &= \mathbb{P}(B) \lim \mathbb{P}( F_n ) \\ &= \mathbb{P}(F)\mathbb{P}(B), \end{align*} so $F \in \mathcal{D}$.

Thus $\mathcal{D}$ is a $\lambda$-system, and since $\pi$ is stable under finite intersections, by the Monotone Class Lemma, $\sigma(\pi) = \mathcal{M}(\pi) \subseteq \mathcal{D}$, the subset relation following from the fact that $\mathcal{M}(\pi)$ is the minimal $\lambda$-system containing $\pi$.

But $\sigma(\pi) = \sigma(X_i : i \in I)$, and thus $\sigma(X_i : i \in I)$ is independent from $\mathcal{G}$ (as anything in $\mathcal{D}$ is independent from $\mathcal{G}$ by definition).

Proof of the other consequence is similar (take $\mathcal{D} := \{ A \in \mathcal{F} : \mathbb{E}\left[Z 1_A \right] = \mathbb{E}\left[\mathbb{E}\left[Z | \sigma(X_{i_0})\right] 1_A \right] \}$).