> [!theorem] > > Let $\mathcal{P}$ be a $\pi$-system over a ground [[Set|set]] $\Omega$, then > $ > \sigma(\mathcal{P}) = \bigcap_{\cf \supset P: \cf\ \sigma\text{-field}}\cf = \bigcap_{\cf \supset \mathcal{P}: \cf\ \lambda\text{-system}}\cf = \lambda(\mathcal{P}) > $ > In other words, the [[Sigma Algebra|sigma algebra]] generated by $\mathcal{P}$ is the same as the [[Lambda-System|lambda-system]] generated by $\mathcal{P}$. > > *Proof*. Since every $\sigma$-field is a $\lambda$-system, $\sigma(\mathcal{P}) \supset \lambda(\mathcal{P})$, so it's sufficient to show that $\lambda(\mathcal{P})$ is a $\sigma$-field as well. As $\lambda(\mathcal{P})$ is a $\lambda$-system, it would be a $\sigma$-field if it is also a $\pi$-system. > > ### Cooperative Sets > > Let > $ > \cm = \bracs{E \in \lambda(\mathcal{P}): E \cap F \in \lambda(\mathcal{P}) \quad \forall F \in \mathcal P} > $ > be the collection of *cooperative* sets, then $\cn$ is a $\lambda$-system with $\cn = \lambda(\mathcal{P})$. > > *Proof*. Let $E, F \in \cm$ such that $F \supset E$, and $G \in \lambda(\mathcal{P})$, then > $ > (F \setminus E) \cap G = \underbrace{(F \cap G)}_{\in \lambda(\mathcal{P})} \setminus \underbrace{(E \cap G)}_{ \in \lambda(\mathcal{P})} \in \lambda(\mathcal{P}) > $ > and $F \setminus E \in \cm$. > > Let $\seq{E_n} \subset \cm$, then > $ > \paren{\bigcup_{n \in \nat}E_n} \cap G = \bigcup_{n \in \nat}\underbrace{(E_n \cap G)}_{\in \lambda(\mathcal{P})} \in \lambda(\mathcal{P}) > $ > and $\bigcup_{n \in \nat}E_n \in \cm$. Therefore $\cm$ is a $\lambda$-system. > > Since $\mathcal{P}$ is a $\pi$-system, $\mathcal{P} \subset \cm$. As $\cm$ is a $\lambda$-system containing $\mathcal{P}$, $\cm = \lambda(\mathcal{P})$. > > > ### Helpful Sets > > Let > $ > \cn = \bracs{E \in \lambda(\mathcal{P}): E \cap F \in \lambda(\mathcal{P}) \quad \forall F \in \lambda(\mathcal{P})} > $ > be the collection of *helpful* sets, then $\cn$ is a $\lambda$-system with $\cn = \lambda(\mathcal{P})$. > > *Proof*. Let $E \in \mathcal{P}$, then since $\cm = \lambda(\mathcal{P})$, $E \cap F \in \lambda(\mathcal{P})$ for all $F \in \cm = \lambda(\mathcal{P})$. Therefore $E \in \cn$. > > By similar arguments as above, $\cn$ is a $\lambda$-system. > > > ### $\pi$-system > > Since $\cn = \lambda(\mathcal{P})$, we have $E \cap F \in \lambda(\mathcal{P})$ for all $E, F \in \lambda(\mathcal{P})$, making it a $\pi$-system.