Unformatted text preview: B , it follows that B ⊆ σ ( D ). ° Exercise 16.4. Prove that a function f : X → R is Mmeasurable if and only if it is Mmeasurable viewed as a function into R . Proof. First suppose f is Mmeasurable 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 Mmeasurable 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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 Spring '09
 Algebra, Sets, Borel, open intervals, School of Mathematical and Statistical Sciences

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