> [!NOTE] Definition 1 (Event Space) > Let $\Omega$ be a [[Definition of a Sample Space|sample space]] and let $\mathcal{F} \subset P(\Omega)$ be a collection of subsets of $\Omega$. $\mathcal{F}$ is an *event space* if it satisfies: > > (ES1) Non-empty: $\Omega \in \mathcal{F}$ > > (ES2) [[Closure under binary operation|Closure under binary operation]] under [[Complement of Set|set compliment]]: for all $A\in \mathcal{F},$ $\Omega \setminus A \in \mathcal{F}$ > > (ES3) Closure under [[Countable Union|countable union]]: for all sequences of events $A_{1},A_{2},\dots \in \mathcal{F},$ $\bigcup_{i=1}^{\infty} A_{i} \in \mathcal{F}.$ > [!NOTE] Definiton 2 (Event Space) > An event space of $\Omega$ is a [[Sigma Algebra|sigma-algebra]] on $\Omega.$ # Properties By [[Event Spaces are Closed Under Finite Unions]], .... By [[Event Spaces are Closed Under Finite Intersections]], ... By [[Event Spaces are Closed under Countably Infinite Intersections]], ... # Applications