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section_1 - Section 1 Probability Spaces Properties of...

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Section 1 Probability Spaces, Properties of Probability. A pair , S ) is a measurable space if S is a ε -algebra of subsets of Π. A collection A of subsets of Π is an algebra (ring) if: 1. Π A. 2. C, B A = C B, C B A. 3. B A = Π \ B A. 4. A is a ε -algebra, if in addition, C i A, i 1 = C i A. i 1 , S , P ) is a probability space if P is a probability measure on S , i.e. 1. P (Π) = 1 . 2. P ( A ) 0 , A ⊂ S . 3. P is countably additive: A i ⊂ S , i 1 , A i A j = ∞ � i = j = P A i = P ( A i ) . i =1 i =1 An equivalent formulation of Property 3 is: 3 . P is a finitely additive measure and B n B n +1 , B n = B = P ( B ) = lim P ( B n ) . n n 1 Lemma 1 Properties 3 and 3 are equivalent. Proof. 1
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3 = 3 : Let C n = B n \ B n +1 , then B n = B C k - all disjoint. k n By 3, P ( B n ) = P ( B ) + P ( C k ) P ( B ) when n ∃ ↓ .
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