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hw6solutions

hw6solutions - Math 318 HW#6 Solutions 1(a Exercise 16.7...

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Unformatted text preview: Math 318 HW #6 Solutions 1. (a) Exercise 16.7. Prove Corollary 10.10, which says that if A 1 ⊃ A 2 ⊃ A 3 ⊃ ··· are mea- surable subsets of E , then T ∞ i =1 A i is measurable and m * ( T ∞ i =1 A i ) = lim i →∞ m * ( A i ). Proof. Let B i = E \ A i for each i . Then B 1 ⊂ B 2 ⊂ B 3 ⊂ ··· and so, by Corollary 10.9, S ∞ i =1 B i is measurable and m * ∞ [ i =1 B i ! = lim i →∞ m * ( B i ) . Therefore, ∞ i =1 A i = ∞ i =1 ( E \ B i ) = E \ ∞ [ i =1 B i is measurable by Corollary 8.4 and m * ∞ i =1 A i ! = 1- m * ∞ [ i =1 B i ! = 1- lim i →∞ m * ( B i ) . But then m * ( B i ) = 1- m * ( A i ) for each i , so we have m * ∞ i =1 A i ! = 1- lim i →∞ (1- m * ( A i )) = lim i →∞ m * ( A i ) , as desired. (b) Exercise 16.36. Show that Corollary 10.10 is false for unbounded sets A i , i = 1 , 2 ,... . Where does the proof of Corollary 10.10 break down in this case?...
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hw6solutions - Math 318 HW#6 Solutions 1(a Exercise 16.7...

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