Theorem

Kolmogorov zero-one law

If $\{X_n\}$ are independent random variables with tail $\sigma$-field $\mathcal{T}$ then $\Lambda \in \mathcal{T}$ implies $\mathbb{P}(\Lambda) \in \{0, 1\}$, so that the tail $\sigma$-field $\mathcal{T}$ is almost trivial.