Let \(\mathcal{A}\) be the smallest \(\sigma\)-algebra of subsets of \(\Omega\) which contains a collections \(\{A_{n},~ n\geq 1\}\) of events. If for every \(n \geq 1\), \(A \in \mathcal{A}\) is independent of \(\{A_{i},~i=1,\dots,n\}\), then either \(\mathbb{P}(A)=0\) or \(\mathbb{P}(A)=1\).
Intuitively, the result states that certain events must have probability 0 or 1 and no intermediate value, for example either something happens infinitely often or it doesn’t.
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