Unformatted text preview: MATH 587 Solutions to Assignment 1 October 12, 2004 (1) Define µ ( A ) = ∑ ω ∈ A p ( ω ) , A ⊂ Ω. Obviously µ ( A ) ≥ 0 for all A . Since µ ( { ω } ) = p ( ω ) < ∞ , then µ is not identically infinite. Finally, let A n , n ≥ 1 be pw disjoint subsets of Ω, and let A = ∪ ∞ n =1 A n . Then µ ( A ) = ∑ ω ∈ A p ( ω ) = ∑ ∞ n =1 ∑ ω ∈ A n p ( ω ) = ∑ ∞ n =1 µ ( A n ). Thus µ is countably additive, so is a measure. µ is σfinite since if we enumerate Ω as Ω = { ω 1 , ω 2 , . . . } and let A n = { ω n } , then Ω = ∪ ∞ n =1 A n where µ ( A n ) = p ( ω n ) < ∞ . (2) Suppose C is a λsystem. If A, B ∈ C with A ⊂ B , then B \ A = B ∩ A c = ( B c ∪ A ) c ∈ C since B c ∩ A = ∅ . Suppose { A n , n ≥ 1 } ⊂ C with A n ⊂ A n +1 ∀ n . Set B 1 = A 1 and B n = A n \ A n − 1 ∀ n ≥ 2. The B n ’s are pairwise disjoint and B n ∈ C∀ n . Therefore ∪ ∞ n =1 A n = ∪ ∞ n =1 B n ∈ C . Hence C is a dsystem.system....
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This note was uploaded on 01/25/2011 for the course STAT 235a at Stanford.
 '07
 RomanVershynin
 Probability

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