M3yQw2-09

M3yQw2-09 - B it follows that B ⊆ σ D ° Exercise 16.4...

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ASSIGNMENT 9 · SOLUTIONS MAT 473 · SPRING 2012 Exercise 16.2. Prove that each of the following collections generates the σ -algebra B of Borel sets in R : (ii) { ( a, b ] | a<b in R } ,and (vii) { [ t, + ) | t R } . Proof. Let C be the collection in part (ii), and let σ ( C )bethe σ -algebra generated by C .F o r any a<b in R , we have ( a, b ]= ° n =1 ( a, b +1 /n ) ∈B , so C⊆B ,anditfo l lows(s ince B is a σ -algebra) that σ ( C ) ⊆B . On the other hand, for each a<b in R , ( a, b )= ± n =1 ( a, b 1 /n ] σ ( C ) , so the collection of all bounded open intervals is contained in σ ( C ), and since the bounded open intervals generate B ,itfo l lowsthat B⊆ σ ( C ). Now let D be the collection in part (vii), and let σ ( D )bethe σ -algebra it generates. For each t R , [ t, + )=( −∞ ,t ) c ∈B , so D⊆B . On the other hand, for each a<b in R ,[ a, b )=[ a, + ) [ b, + ) c σ ( D ), so ( a, b )= ± n =1 [ a +1 /n, b )
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Unformatted text preview: B , it follows that B ⊆ σ ( D ). ° Exercise 16.4. Prove that a function f : X → R is M-measurable if and only if it is M-measurable viewed as a function into R . Proof. First suppose f is M-measurable into R , and B ∈ B . Then B = C ∪ D , where C ∈ B and D ⊆ {−∞ , + ∞} . Thus f − 1 ( B ) = f − 1 ( C ∪ D ) = f − 1 ( C ) ∪ f − 1 ( D ) = f − 1 ( C ) ∪∅ = f − 1 ( C ) , and f − 1 ( C ) ∈ M by assumption. Thus f is M-measurable into R . Since B ⊆ B , the converse is immediate. ° Date : March 8, 2012. S. Kaliszewski, School of Mathematical and Statistical Sciences, Arizona State University. 1...
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