Probability space

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 .