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Sigma Alpha

xAΩα2Ωx \in A \subset \Omega \in \alpha \subset 2^{\Omega}

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

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

PP, α[0, 1]\alpha \rightarrow [0,~1] and CA (Countably Additive)

P(k=1Ak)=k=1P(Ak)P(\cup_{k=1}^{\infty} A_k) = \sum\limits_{k=1}^{\infty}P(A_k) if A1Aj=, ij, P(Ω)=1A_1 \cap A_j = \emptyset,~\forall i \neq j,~P(\Omega) = 1

(P, α, Ω)(P,~\alpha,~\Omega) is the probability space.

AA and BB are mutually exclusive.

AB=A \cap B = \emptyset

AA and BB are independent

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

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

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