> [!definition] > > Let $\Omega$ be a [[Set|set]] and $\alg \subset \pow{\Omega}$ be a collection of subsets with $\emptyset \in \alg$, then $\alg$ is a **ring** if: > - $E, F \in A$ implies that $E \cup F \in \alg$, > - $E, F \in \alg$ implies that $F \setminus E \in \alg$. > > and $\alg$ is a $\pi$**-system** if > - $E, F \in \alg$ implies that $E \cap F \in \alg$ > > and $\alg$ is a **field** if $\alg$ is a ring and > - $E \in \alg$ implies that $E^c \in \alg$. > > and lastly, $\alg$ is a $\sigma$-field ([[Sigma Algebra|sigma algebra]]) if it is a field that is closed under countable unions.