Codex @codex  0

Dynkin lemma, also called the pi-lambda theorem, states that if a Dynkin system $\mathcal{D}$ contains a pi-system $\mathcal{P}$, then

Let $\lambda (\mathcal{P})$ be the smallest Dynkin system containing $\mathcal{P}$. It is enough to prove that $\lambda (\mathcal{P})$ is a sigma-algebra, because then

First fix $A\in \mathcal{P}$ and defineThis is a Dynkin system. It contains $\Omega$ because $A\in \lambda (\mathcal{P})$; closure under relative complements follows fromand closure under disjoint unions follows by distributing $A\cap -$ over the union. Since $\mathcal{P}$ is closed under intersections, $\mathcal{P}\subseteq {\mathcal{G}}_A$. Minimality therefore givesThus $A\cap B\in \lambda (\mathcal{P})$ whenever $A\in \mathcal{P}$ and $B\in \lambda (\mathcal{P})$.

Now fix $B\in \lambda (\mathcal{P})$ and defineThe same argument shows that ${\mathcal{H}}_B$ is a Dynkin system, and the preceding paragraph shows that it contains $\mathcal{P}$. Hence ${\mathcal{H}}_B=\lambda (\mathcal{P})$. We have proved that $\lambda (\mathcal{P})$ is itself a pi-system.

A Dynkin system that is also a pi-system is a sigma-algebra: it is closed under arbitrary finite intersections, hence finite unions by complements, and any countable union can be disjointified before using closure under disjoint unions. Therefore $\lambda (\mathcal{P})=\sigma (\mathcal{P})$, proving the lemma.

Fix $A_1\in {\mathcal{A}}_1$ and letThis is a Dynkin system. It contains $\Omega$. If $B\in {\mathcal{D}}_{A_1}$, thenso it is closed under complements. Countable additivity gives closure under countable disjoint unions.

The hypothesis says ${\mathcal{A}}_2\subseteq {\mathcal{D}}_{A_1}$. Since ${\mathcal{A}}_2$ is a pi-system, Dynkin lemma yieldsThus the factorization holds for every $A_1\in {\mathcal{A}}_1$ and every $B\in \sigma ({\mathcal{A}}_2)$.

Now fix such a $B$ and defineThe identical calculation makes ${\mathcal{E}}_B$ a Dynkin system. The first step gives ${\mathcal{A}}_1\subseteq {\mathcal{E}}_B$, so a second application of Dynkin lemma givesTherefore the probability factorization holds for everywhich is precisely independence of the two sigma-algebras. This is independence extended from generating pi-systems.

Let $\mathcal{G}=\sigma (A_1,A_2,\dots )$. The limit $Y$ from part (c) is $\mathcal{G}$-measurable because each $X_n$ is. If $A\in {\bigcup }_n{\mathcal{F}}_n$, then $A\in {\mathcal{F}}_N$ for some $N$, and for every $n\ge N$,The $L^1$ convergence lets us pass to the limit and obtain $\mathbb{E}[Y{1}_A]=\mathbb{E}[X{1}_A]$.

The sets on which this identity holds form a Dynkin system, and ${\bigcup }_n{\mathcal{F}}_n$ is a generating pi-system. The Dynkin lemma therefore extends the identity to every $A\in \mathcal{G}$. Consequently $Y$ has both defining properties of $\mathbb{E}[X\mid \mathcal{G}]$, established here without appealing to the general existence theorem.

If a Dynkin system contains a pi-system, then it contains the sigma-algebra generated by that pi-system. This proves uniqueness of finite measures from agreement on a generating pi-system.