Why for independent events, the necessary condition for $P(\limsup A_n)>0$ is sufficient for $P(\limsup A_n)=1$?
2026-03-27 11:47:50.1774612070
Why is $P(\limsup A_n)>0$ sufficient for $P(\limsup A_n)=1$?
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Because, in full generality, $\limsup A_n$ belongs to the tail sigma-algebra generated by the sequence $(A_n)$ and because, in the present case, the sequence $(A_n)$ is independent hence, according to Kolmogorov's zero-one law, the tail sigma-algebra is trivial in the sense that every event in it has probability $0$ or $1$.