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Unformatted text preview: Section 1 Probability Spaces, Properties of Probability. A pair (Π , S ) is a measurable space if S is a εalgebra 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 εalgebra, if in addition, C i ⊂ A, i ∗ 1 = ≥ C i ⊂ A. i ∗ 1 (Π , S , P ) is a probability space if P is a probability measure on S , i.e. 1. P (Π) = 1 . 2. P ( A ) ∗ , 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 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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This note was uploaded on 10/28/2008 for the course MATH 18.05 taught by Professor Panchenko during the Spring '05 term at MIT.
 Spring '05
 Panchenko
 Statistics, Algebra, Sets, Probability

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