> [!definition] > > Let $\Omega$ be a [[Set|set]] and $\alg \subset \pow\Omega$ be a collection of subsets. We say that $\alg$ is a $\lambda$-system if > - The whole space $\Omega \in \alg$. > - For any $E, F \subset \alg$ where $E \subset F$, we have $F \setminus E \in \alg$. > - For any increasing sequence $\seq{A_n} \subset \alg$, $\bigcup_{n \in \nat}A_n \in \alg$. > [!theorem] > > Let $\Omega$ be a ground set. If $\alg$ is a $\pi$-system *and* a $\lambda$-system, then $\alg$ is a [[Sigma Algebra|sigma algebra]]. > > *Proof*. Since $\Omega \in \alg$, $\alg$ is closed under complements. > > If a $\lambda$-system is closed under finite unions, then it is also closed under countable unions, so we can reduce this to simply finite unions. To this end, just use complements to write unions as intersections. For any $E, F \in \alg$, > $ > E \cup F = (E \cup F)^{cc} = (E^c \cap F^c)^c > $