# Questions on Kolmogorov Zero-One Law Proof in Rosenthal

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Rosenthal's Probability book:

Here are my questions:

Question 1: In the first red box, does the fact that Q and P agree on J hold because of the statement in the blue box?

Question 2: What is the relevance of "independence is defined in terms of finite subcollections only" ? I was thinking that we can infer the independence of $A, A_1, A_2, ...$ merely from the $\forall n \in \mathbb{N}$.

Here is what I think "independence is defined in terms of finite subcollections only" means:

Given events $(A_n)_{n \geq 1}$, they are defined to be independent if for any indices $i_1, i_2, ..., i_n$,

$P(\bigcap_{k = i_1}^{i_n} A_{k}) = \prod_{k = i_1}^{i_n} P(A_{k})$.

Thus, the "finite subcollections" refers to the $A_{i_1}, A_{i_2}, ..., A_{i_n}$

Assuming I understood that right, how that is relevant?

• For Question 1, the answer is "Yes". As for Question 2, what you say is OK; and it is relevant because any finite subcollection of $A, A_1, A_2,\dots$ is contained in $A,A_1, \dots ,A_{n-1}$ for some $n$. – Etienne May 24 '15 at 17:01
• @Etienne If so, how exactly can we conclude such? I sort of get it intuitively, I think but am unable to express it precisely. My guess: If $A, A_1, A_2, ..., A_{n-1}$ are independent, then any finite subcollection of that is independent. If this indeed holds $\forall n \in \mathbb{N}$, then finite subcollections of $A, A_1, A_2, ...$ are also independent. Hence, $A, A_1, A_2, ...$ is independent ? – BCLC May 24 '15 at 18:13
• Yes, this is it. – Etienne May 24 '15 at 21:05
• @Etienne Thanks! – BCLC May 28 '15 at 9:11
• You're welcome! – Etienne May 29 '15 at 16:46