Pi-System

A $\pi$-system $\mathcal{P}$ is a Family of Sets satisfying the following property

• $\forall A,B\in \mathcal{P},A\cap B\in \mathcal{P}$

Examples

The all below are the pi-system on $\Omega =\mathbb{R}$

• $\mathcal{A}=\{(a,b]:-\infty <a<b<\infty \}\cup \{\varnothing \}$
• $\mathcal{A}=\{[a,b):-\infty <a<b<\infty \}\cup \{\varnothing \}$
• $\mathcal{A}=\{(a,b):-\infty <a<b<\infty \}\cup \{\varnothing \}$
• $\mathcal{A}=\{[a,b]:-\infty <a<b<\infty \}\cup \{\varnothing \}$