Definition

probability space

a triple $(\Omega, \mathcal{B}, \mathbb{P})$ where

• $\Omega$ is the sample space
• $\mathcal{B}$ is a $\sigma$-algebra of subsets of $\Omega$
• $\mathbb{P}$ is a probability measure.

A probability measure is a function $\mathbb{P}: \mathcal{B} \to [0,1]$ such that

1. $\mathbb{P}(A) \geq 0$ for all $A \in \mathcal{B}$
2. $\mathbb{P}$ is $\sigma$-additive: if $\{A_n: n \geq 1\}$ are events in $\mathcal{B}$ that are disjoint then $\mathbb{P}\left(\cup_{n=1}^{\infty}A_n\right) = \Sigma_{n=1}^{\infty}\mathbb{P}(A_n)$
3. $\mathbb{P}(\Omega) = 1$.