I'm reading Durrett's Probability Theory and Examples (4th ed.) and I am having a hard time understanding the proof of Theorem 2.1.5:
Suppose $\mathcal{F_{i,j}}$, $1\leq i \leq n$, $1\leq j \leq m(i)$ are independent and let $\mathcal{G_{i}}=\sigma(\bigcup_{j}\mathcal{F_{i,j}})$. Then $\mathcal{G_{1}},…,\mathcal{G_{n}}$ are independent.
Proof in Durrett:
Let $\mathcal{A_{i}}$ be the collection of sets of the form $\bigcap_{j}A_{i,j}$ where ${A_{i,j}}\in\mathcal{F_{i,j}}$. $\mathcal{A_{i}}$ is a $\pi$-system that contains $\Omega$ and contains $\bigcup_{j}\mathcal{F_{i,j}}$ so theorem 2.1.3 implies that $\sigma(\mathcal{A_{i}})=\mathcal{G_{i}}$ are independent.
I don't understand a couple of things in this proof:
i) why does $\mathcal{A_{i}}$ contain $\bigcup_{j}\mathcal{F_{i,j}}$ ? I've tried proving that an element of $\bigcup_{j}\mathcal{F_{i,j}}$ must be in $\mathcal{A_{i}}$, but to no avail as of yet.
ii) even if I prove that $\bigcup_{j}\mathcal{F_{i,j}}\subset \mathcal{A_{i}}$ does the fact then somehow imply that $\sigma(\mathcal{A_{i}})=\sigma(\bigcup_{j}\mathcal{F_{i,j}})=\mathcal{G_{i}}$ ?
I've been struggling with this for a while and have run out of ideas on how to prove this or what Durrett is doing here. Any insight would be much appreciated!
Best Answer
For (1), for some sets $A_i \in \cup_{j} \mathcal{F_{i,j}}$, there exists $j$ such that $A_i\in \mathcal{F_{i,j}}$. Denote such $A_i$ as $A_{i,j}$. So $A_{i,j}\in \cap_{j} A_{i,j}$.
For (2), if you want to show two sigma-algebra are equal that is $\sigma(\mathcal{A}_{i})=\sigma(\cup_j\mathcal{F_{i,j}})$. It suffices to prove $\mathcal{A}_{i}\subset \sigma(\cup_j\mathcal{F_{i,j}})$ and $\cup_j\mathcal{F_{i,j}}\subset \sigma(\mathcal{A}_{i})$. Note that $\cup_{j}\mathcal{F_{i,j}}\subset \mathcal{A}_{i}$ then $\sigma(\cup_{j}\mathcal{F_{i,j}})\subset \sigma(\mathcal{A}_{i})$. Conversely, it is clear that $\mathcal{A}_{i}\subset \sigma(\cup_j\mathcal{F_{i,j}})$.