Proof of Grouping Lemma for Independent Random Variables

By Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras, we must show: for every finite nonempty set of distinct indices $b_1,\dots,b_p\in B$ and all events $A_l\in\mathcal{G}_{b_l}$,

Throughout, $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra and products use the finite product notation.

Step 1 (a generating $\pi$-system for each block). For $b\in B$ let $\mathcal{P}_b$ be the family of all sets of the form $\bigcap_{m\in E}X_m^{-1}(C_m)$, where $E\subseteq I_b$ is finite and nonempty and $C_m\in\mathcal{B}(\mathbb{R})$ for each $m\in E$. The intersection of two such sets is again of this form (intersect over the union of the two index sets, taking $C_m=\mathbb{R}$ for indices missing from one of them, and replacing $C_m$ by the intersection of the two Borel sets on shared indices), so $\mathcal{P}_b$ is a $\pi$-system. Note $\Omega\in\mathcal{P}_b$: take $E=\{m\}$ for any $m\in I_b$ and $C_m=\mathbb{R}$, so that $X_m^{-1}(\mathbb{R})=\Omega$. The family $\mathcal{P}_b$ contains the generating family $\mathcal{C}_b=\{X_m^{-1}(C):m\in I_b,\ C\in\mathcal{B}(\mathbb{R})\}$ of $\mathcal{G}_b$, and every member of $\mathcal{P}_b$ lies in $\mathcal{G}_b$ because $\sigma$-algebras are closed under finite intersections. By minimality of the generated $\sigma$-algebra, $\mathcal{G}_b=\sigma(\mathcal{C}_b)\subseteq\sigma(\mathcal{P}_b)\subseteq\mathcal{G}_b$, so $\sigma(\mathcal{P}_b)=\mathcal{G}_b$.

Step 2 (the product formula on the $\pi$-systems). Fix distinct $b_1,\dots,b_p\in B$ and sets $A_l=\bigcap_{m\in E_l}X_m^{-1}(C_m)\in\mathcal{P}_{b_l}$. The index sets $E_1,\dots,E_p$ are pairwise disjoint because $I_{b_1},\dots,I_{b_p}$ are. Let $E=E_1\cup\dots\cup E_p$; this is a finite set of indices, and the subfamily $(X_m)_{m\in E}$ is independent because, by Independence of Events and of Random Variables, every finite subfamily of an independent family is independent. Applying the defining product formula of Independence of Events and of Random Variables to the events $X_m^{-1}(C_m)$, $m\in E$, and also separately to each subfamily $(X_m)_{m\in E_l}$, gives

where the regrouping of the finite product over $E$ into the iterated product is the associativity and commutativity of finite products of real numbers.

Step 3 (extension from $\pi$-systems to the $\sigma$-algebras). We claim: for $0\le k\le p$, the identity ($*$) holds whenever $A_l\in\mathcal{G}_{b_l}$ for $l\le k$ and $A_l\in\mathcal{P}_{b_l}$ for $l>k$. The case $k=0$ is Step 2. Assume the claim for $k-1$ and fix $A_l\in\mathcal{G}_{b_l}$ for $l<k$ and $A_l\in\mathcal{P}_{b_l}$ for $l>k$. Let

where for $p=1$ the intersection over the empty index set is read as $\Omega$ and the empty product as $1$. We check that $\mathcal{L}$ is a $\lambda$-system. Write $D=\bigcap_{l\ne k}A_l$ and $c=\prod_{l\ne k}P(A_l)$. (1) $\Omega\in\mathcal{L}$: since $\Omega\in\mathcal{P}_{b_k}$ (Step 1), the induction hypothesis for $k-1$ applied with the choice $A_k=\Omega$ gives $P(\Omega\cap D)=P(\Omega)\,c=c$, using $P(\Omega)=1$ from the definition of a probability measure; and $P(\Omega)\,c$ is exactly the required right-hand side for $A=\Omega$. (2) If $A\subseteq A'$ both lie in $\mathcal{L}$, then by finite additivity of the measure $P$ applied to the disjoint union $A'\cap D=(A\cap D)\cup((A'\setminus A)\cap D)$,

using $P(A')=P(A)+P(A'\setminus A)$; so $A'\setminus A\in\mathcal{L}$. (3) If $(A^{(m)})_{m\in\mathbb{N}}$ is a nondecreasing sequence in $\mathcal{L}$ with union $A$, write $A$ as the disjoint union of the sets $A^{(1)}$ and $A^{(m+1)}\setminus A^{(m)}$, $m\ge1$; countable additivity of $P$ gives $P(A\cap D)=\lim_m P(A^{(m)}\cap D)$ and $P(A)=\lim_m P(A^{(m)})$ (the partial sums of the two series are exactly $P(A^{(m)}\cap D)$ and $P(A^{(m)})$ by finite additivity), hence $P(A\cap D)=\lim_m P(A^{(m)})c=P(A)c$ by the algebra of limits; so $A\in\mathcal{L}$.

By the induction hypothesis for $k-1$ (with $A_k$ ranging over $\mathcal{P}_{b_k}$), $\mathcal{P}_{b_k}\subseteq\mathcal{L}$. By Dynkin's Pi-Lambda Theorem and Step 1, $\mathcal{G}_{b_k}=\sigma(\mathcal{P}_{b_k})\subseteq\mathcal{L}$, which is precisely the claim for $k$. Taking $k=p$ proves ($*$) for all $A_l\in\mathcal{G}_{b_l}$, so the family $(\mathcal{G}_b)_{b\in B}$ is independent.

Step 4 (independent block-measurable random variables). Let $Y_b$ be $\mathcal{G}_b$-measurable for each $b\in B$. By Independence of Events and of Random Variables it suffices to show that every finite subfamily $Y_{b_1},\dots,Y_{b_p}$ (distinct $b_l$) is independent, i.e., that for all Borel sets $C_1,\dots,C_p$ the events $Y_{b_l}^{-1}(C_l)$ are independent. For every nonempty $S\subseteq\{1,\dots,p\}$, the events $Y_{b_l}^{-1}(C_l)\in\mathcal{G}_{b_l}$, $l\in S$, are chosen from $\sigma$-algebras with distinct indices, so ($*$) gives

which is exactly the required product formula for every subfamily. Hence $(Y_b)_{b\in B}$ is independent. $\blacksquare$