Countable Intersection of Events is Event

Theorem

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.

The event space $\Sigma$ of $\EE$ has the property that:

$\quad A_1, A_2, \ldots \in \Sigma \implies \ds \bigcap_{i \mathop = 1}^\infty A_i \in \Sigma$

That is, the countable intersection of events is also an event.

Proof

By definition, a probability space $\struct {\Omega, \Sigma, \Pr}$ is a measure space.

So, again by definition, an event space $\Sigma$ is a $\sigma$-algebra on $\Omega$.

From Sigma-Algebra is Delta-Algebra:

$\ds A_1, A_2, \ldots \in \Sigma \implies \bigcap_{i \mathop = 1}^\infty A_i \in \Sigma$

by definition of $\delta$-algebra.

$\blacksquare$

Also see

• Elementary Properties of Event Space

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.4$: Probability spaces: $(13)$