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HW1-320-A

# HW1-320-A - B is a Borel set show that f B is also a Borel...

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Math 320 Measure Theory and Integration Assignment 1: ° -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0 ; 1] which admit decimal expansions such that the digits 2 ; 4 ; 6 ; 8 all appear at least once. (Thus, 0.0032004068 is an example while 0.232046 is not. For 0.9999 .... , write 1 instead.) Show that A is a Borel set. 2. Given sets X and Y . Let f : X ! Y be any function. (a) Show that if A is a ° -algebra of subsets of Y , then f ° 1 ( A ) is a ° -algebra of subsets of X . (b) Show that ° ( f ° 1 ( S )) = f ° 1 ( ° ( S )) , for any collection S of subsets of Y . Here ° ( K ) represents the ° -algebra generated by K . 3. Let f be a strictly increasing continuous real-valued function on R . If
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Unformatted text preview: B is a Borel set, show that f ( B ) is also a Borel set. 4. Let f : R ! R . (a) Suppose f is a monotone function, show that the set of points at which f is discontinuous is countable. (b) Suppose f is any function. Show that the set of points at which f is discontinuous is a Borel set. 5. (a) Prove the Baire Category Theorem: If U 1 , U 2 ,... are open dense subsets of R n , then T 1 i =1 U i is also dense (it may not be open). A subset U of R n is said to be dense if O \ U 6 = ; for every non-empty open set O & R n . (b) Find a Borel set in R that is neither an F & nor a G ± . 1...
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