Probability space - Knowlpedia

A probability space is a triple $(\Omega,\mathcal{F},\mathbb{P})$ where $\Omega$ is a set, $\mathcal{F}$ is a sigma-algebra on $\Omega$ , and $\mathbb{P}$ is a probability measure on $(\Omega,\mathcal{F})$ .

Elements of $\mathcal{F}$ are the events whose probabilities are evaluated by $\mathbb{P}$ , and a random variable is a measurable function defined on $(\Omega,\mathcal{F},\mathbb{P})$ .

Examples

Fair coin toss: $\Omega=\{H,T\}$ , $\mathcal{F}=2^\Omega$ , and $\mathbb{P}(\{H\})=\mathbb{P}(\{T\})=1/2$ .
Uniform draw from $[0,1]$ : $\Omega=[0,1]$ , $\mathcal{F}$ is the Borel $\sigma$ -algebra, and $\mathbb{P}(A)=\lambda(A)$ where $\lambda$ is Lebesgue measure.