No1(2004)solutions

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

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

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. Conversely, suppose C is a d -system. If A ∈ C , then A c = Ω \ A ∈ C . If A, B ∈ C with A B = , then B A c so A B = ( A c \ B ) c ∈ C . Suppose { A n , n 1 } ⊂ C are pairwise disjoint. Set
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern