What is an example of a pi-system that is not a sigma algebra?

Question

What is an example of a pi-system that is not a sigma algebra?

We already have: What is an example of a lambda-system that is not a sigma algebra?

Answer 1

Consider the pi-system $P = \{\{\}, \{1\}, \{1, 2\}, \{1, 3\}\}$ on $\Omega = \{1, 2, 3\}$. It is easy to verify that it is indeed a $\pi$-system by taking the intersection of any two sets. On the other hand, it cannot be a sigma algebra since $\{1, 2\} \cup \{1, 3\} = \{1, 2, 3\} \not\in P$.

Answer 2

The collection of half rays $(a,+\infty)$ with $a\in\Bbb{R}$ is stable finite intersection, so it's a $\pi$-system. However, it's not a $\sigma$-algebra since it's complement $(-\infty,a]$ is not included.