ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4329 on D5500: Almost sure event
Probability of countable intersection with an almost sure event

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$ (ii) $$\mathbb{P}(E_0) = 1$$
Then $$\mathbb{P} \left( \bigcap_{n = 0}^{\infty} E_n \right) = \mathbb{P} \left( \bigcap_{n = 1}^{\infty} E_n \right)$$
Subresults
 ▶ R4330: Probability of finite intersection with an almost sure event
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$ (ii) $$\mathbb{P}(E_0) = 1$$
Since we can write $$\bigcap_{n = 0}^{\infty} E_n = E_0 \cap \left( \bigcap_{n = 1}^{\infty} E_n \right) = E_0 \cap F$$ with $F : = \bigcap_{n = 1}^{\infty} E_n$, then this result is a consequence of R4331: Probability of binary intersection with an almost sure event. $\square$