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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Some confusion may come from considering the product space $\mathbb{R}^\mathbb{Z}$, rectangles, and projections as central to the problem. They are not.

Let's go back to the beginning. Your first sentence says "Let $X_i$ be a sequence of i.i.d. rv.". This means there is a probability space $(\Omega,{\cal F},\mathbb{P})$ and measurable maps $X_i:(\Omega,{\cal F})\to (\mathbb{R},{\cal B}(\mathbb{R}))$ for $i=1,2,3\dots.$ These maps have other important properties, crucial to the proof of Kolmogorov's $0-1$ theorem, but let's ignore those for now.

You have correctly explained that $B_0:=\sigma(X_1,\ldots)$ is the smallest $\sigma$-algebra that makes all the maps $(X_i)_{i\geq 1}$ measurable. Note that $B_0$ is a subset of $\cal F$. Similarly, $\sigma(X_1,\ldots,X_k)$ is the smallest $\sigma$-algebra that makes all the maps $(X_i)_{1\leq i\leq k}$ measurable, and $B_k$ is the smallest $\sigma$-algebra that makes all the maps $(X_i)_{i\geq k+1}$ measurable. All of these are subsets of $\cal F$; in fact they are all subsets of $B_0$.

1. "After all, it wouldn't make any sense to even consider $\cap B_k$ right?" Sure, why not? Each $B_k$ is a subset of $\cal F$, and $\cap B_k$ means their intersection.
2. "What is the formal definition of $B_k$?" Answer: $B_k$ is the smallest $\sigma$-algebra that makes all the maps $(X_i)_{i\geq k+1}$ measurable. This is the formal definition.
3. "Is $B_k$ a projection?" No, $B_k$ is a subset of $\cal F$.
4. "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?" Answer: $B_k$ is the smallest $\sigma$-algebra that makes all the maps $(X_i)_{i\geq k+1}$ measurable. Nothing needs to be added to it.