Event

Remarks

When $\Omega$ is finite or countable, one usually takes $\mathcal{F}$ to be the power set of $\Omega$. Then each subset of $\Omega$ is an event. For uncountable $\Omega$, such as $\Omega = \mathbb{R}$, one cannot in general give a probability to each subset. So only the sets in $\mathcal{F}$ are events. The set operations have probabilistic names: $A \cup B$ is "$A$ or $B$", $A \cap B$ is "$A$ and $B$", and $\Omega \setminus A$ is "not $A$". The empty set is the impossible event, and $\Omega$ is the sure event. Two events $A$ and $B$ are mutually exclusive when $A \cap B = \varnothing$.