Sigma Alpha

$x \in A \subset \Omega \in \alpha \subset 2^{\Omega}$

$\alpha$ is Sigma Alpha if and only if it is CUT

$(\Omega,~\alpha)$ is the measurable space.

$P$, $\alpha \rightarrow [0,~1]$ and CA (Countably Additive)

$P(\cup_{k=1}^{\infty} A_k) = \sum\limits_{k=1}^{\infty}P(A_k)$ if $A_1 \cap A_j = \emptyset,~\forall i \neq j,~P(\Omega) = 1$

$(P,\alpha,\Omega)$ is the probability space.

$A$ and $B$ are mutually exclusive.

$A \cap B = \emptyset$

$A$ and $B$ are independent

$P(A \cap B) = P(A) P(B)$

$P (A \cup B) = P (A) + P(B) - P (A \cap B)$