Probability of an event

Because events are measurable sets and $\mathbb P$ is a probability measure, event probabilities are countably additive on pairwise disjoint events, with $\mathbb P(\varnothing)=0$ and $\mathbb P(\Omega)=1$.

• In a fair coin-toss space, the event $A=\{H\}$ has probability $\mathbb P(A)=1/2$.
• Under the uniform probability measure on $[0,1]$, the event $A=[0,1/2]$ has probability $\mathbb P(A)=1/2$.