No1(2004)solutions

No1(2004)solutions - MATH 587 Solutions to Assignment 1(1...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 σ-finite 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 λ-system. 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 d-system.-system....
View Full Document

This note was uploaded on 01/25/2011 for the course STAT 235a at Stanford.

Ask a homework question - tutors are online