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Let $X_i$ be a sequence of i.i.d. rv. When talking about $\sigma(X_1,\ldots)$, I understand this to mean the smallest sigma algebra under which $X_i$ are measurable. Formally, we can take $\mathbb{R}^\mathbb{N}$ and then look appropriate subsets of rectangles that generate $\sigma(X_1,\ldots)$. Now, suppose I'm trying to prove say, Kolmogorov 0-1 law. Now textbooks are talking about $\sigma(X_1,\ldots,X_k)$ and $B_k:=\sigma(X_{k+1},\ldots)$. Already I am confused. Surely what is tacit here is that these are some sort of cylinder projections from $\sigma(X_1,\ldots)$. After all, it wouldn't make any sense to even consider $\cap B_k$ right?

Question 1: What is the formal definition of $B_k$. Is it a projection? In other words, do I just consider the smallest sigma algebra generated by $X_{k+1},\ldots$ and then slap on the necessary extra stuff on the left?

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