Probability Space

Probability space is a three tuple
$\{ S, \Sigma, \mathbb{P} \}$

where $S$ is sample space, $\Sigma\subseteq 2^S$ is sigma algebra, and $\mathbb{P}: \Sigma \to [0, 1]$ is the probability function such that,

1. $\mathbb{P}(\emptyset) = 0$
2. Given $A_1, A_2, ... \in \Sigma$, if $A_1, A_2, ...$ are mutually exclusive ( $\forall_{i\neq j} A_i \cap A_j = \emptyset$ ) then $\mathbb{P}( \cup_{n=0}^\infty A_n) = \sum_{n=0}^\infty \mathbb{P}(A_n)$

Definition We say two events A and B are independent if $\mathbb{P}(A\cap B) = \mathbb{P}(A)\mathbb{P}(B)$

This notion of independence is really the one thing that makes probability a different field of study from analysis. Note in general, $\mathbb{P}(A\cap B) = \mathbb{P}(A)\mathbb{P}(B)$ is not something as a result of a proof, it is usually assumed as premise of a problem, because independence gives probability many useful and convenient properties.