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# hw1 - Math 245A Homework 1 Brett Hemenway 3 a Let M be an...

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Unformatted text preview: Math 245A Homework 1 Brett Hemenway October 19, 2005 3. a. Let M be an infinite σ-algebra on a set X Let B x be the intersection of all E ∈ M with x ∈ E . Suppose y ∈ B x , then B y ⊂ B x because B x is a set in M which contains y . But B x ⊂ B y because if y ∈ E for any E ∈ M , then x ∈ E , otherwise y 6∈ B x = B x ∩ E c . So B x = B y . This means that for all x, y , either B x = B y or B x ∩ B y = ∅ . The collection B x x ∈ X must be infinite since by hypothesis M was infinite, and every element E of M either contains B x or is disjoint from it for each x . So the B x form an infinite collection of disjoint sets in M . b. If there are uncountably many different B x ∈ M we are done. Otherwise there are countably many, i.e. we have a sequence { B x n } ∞ n =1 where B x i ∩ B x j = ∅ when i 6 = j . The set { , 1 } ∞ = { , 1 }×{ , 1 }× , . . . is uncountable, and we have an injection { , 1 } ∞ , → M { a i } ∞ i =1 7→ ∞ [ i =1 B x i if a i = 1 ∅ if a i = 0 So M is uncountable. 4. Let A an algebra, suppose A is closed under countable increasing unions. then we wish to show that A is closed under arbitrary count- able unions. Let { E n } ∞ n =1 be a sequence of sets in A . Let F n = ∪ n m =1 E n . Then { F n } ∞ n =1 is an increasing sequence of functions and ∪ N n =1 E n = ∪ N n =1 F n . By our hypothesis, ∪ ∞ n =1 F n ∈ A , so ∪ ∞ n =1 E n ∈ A . The converse, that if A is a σ-algebra it is closed under countable increasing unions, follows from the definition of a σ-algebra....
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hw1 - Math 245A Homework 1 Brett Hemenway 3 a Let M be an...

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