HW1-320-A

HW1-320-A - B is a Borel set, show that f ( B ) is also a...

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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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This note was uploaded on 08/04/2011 for the course MATH 320 taught by Professor Gregorymargulis during the Fall '10 term at Yale.

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