Kolmogorov‘s 0-1 Law seems to say something very intuitive: Tail events of independent objects have probability $0$ or $1$.
At least it should be intuitive - but it is not to me. Why exactly are tail events supposed to have this behaviour? What‘s the intuitive argument?
Intuition is right there in the proof of the theorem. If $A \in \sigma (X_n,X_{n+1},...)$ for every $n$ then $A$ is independent of $X_1,X_2,..,X_n$ for every $n$. So what does $A$ depend on? Intuitively it is independent of itself and only trivial events can be independent of themselves.