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Before starting I should note that I've already read this, this, this and this, and none of them quite provide the intuition I seem to be missing. I've tried to provide as many details related to my own thought process to see if we can diagnose this misunderstanding more specifically.

I just started learning about Kolmogorov's zero-one law, and (not unlike many people who first encounter it I suppose), I'm having trouble wrapping my head around the concept of a tail field. In particular, if we have a sequence of events $A_n\in\mathcal{F}$, then the tail field $\tau$ is defined as $$ \tau = \bigcap_{n=1}^\infty\sigma(A_n, A_{n+1},\cdots) $$ First, I'm slightly confused as to how $\tau$ is a $\sigma$-algebra at all (though I think I can check this myself).

What I'm really struggling with is simply how to interpret this definition. To me, it seems like $\tau$ should be empty, even though I know it's clearly not as $\limsup_n A_n\in\tau$ (and I understand the proof behind why).

As far as I understand, if $A\in\tau$, then $A\in\sigma(A_n,A_{n+1},\cdots)$ for every $n\in\mathbb{N}$. So any $A\in\tau$ can't depend on the first $A_1,\cdots, A_n$ events for any $n$. The problem is that I have no intuition for what kinds of sets behave this way. It seems to me like if you have any formula for $A$ involving the sets $A_n$, then there has to be a minimum $n_0$ for which $A_{n_0}$ is involved in the formula for $A$, but this means $A\not\in\sigma(A_{n_0+1},A_{n_0+2},\cdots)$, meaning $A\in\tau$. So, in my head, $\tau$ can't contain any set that can be written in terms of the $A_n$'s.

As I said above, I know this is false, as $\limsup_n A_n$ and $\liminf_n A_n$ are in $\tau$. To be honest, the more I think about it the less sense the definitions of these two sets seem to make to me as well. I mean, what does $\limsup_n A_n$ even look like if it supposedly doesn't depend on any of the $A_n$'s? If it doesn't depend on any of the $A_n$'s, then $\limsup_n A_n = \limsup_n B_n$ for any two sequences of events $A_n$ and $B_n$, no?

I guess the flaw in my logic has to do with the idea of "dependence". I know there are events in $\limsup_n A_n$, and the logical proposition giving the inclusion of an element in $\limsup_n A_n$ does depend on the $A_n$'s, so what I said above makes no sense (obviously).

But something is still wrong in my head, and I want to figure out how to right it. Can anyone see where the flaw in my logic is?

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Regarding your first question: one can show that the intersection of $\sigma$-algebras is also a $\sigma$-algebra by directly checking the definitions.

Regarding your broader questions, it may help to think about the analogy of a convergent sequence of real numbers: the limit does not depend on any individual element (or finite subset) of the sequence, but we definitely do not have $\lim_n a_n = \lim_n b_n$ in general for any two arbitrary convergent sequences $a_n$ and $b_n$.

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  • $\begingroup$ Ahhhhh, that limit analogy does help a bit. Let me think about it a bit more before accepting your answer right away, I might have a followup question. $\endgroup$ – user3002473 Oct 25 '18 at 20:20

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