hmwk11

# hmwk11 - →[0 1 be the Cantor-Lebesgue function(see SS...

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MATH 202A HOMEWORK 11 The following assignment is due Wed., Nov. 14. SS Ch.1, p. 38, Exercise # 2(b)(c)(d). SS Ch.1, p. 42, Exercise #18. SS Ch.1, p. 43, Exercise #22. SS Ch.1, p. 43, Exercise #23. SS Ch.1, p. 44-45, Exercise #32. Extra Problem . (a) Prove that the preimage of a Borel set under a continuous map is Borel. (b) Construct a Lebesgue measurable set that is not Borel, as follows: Let f : [0 , 1]
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Unformatted text preview: → [0 , 1] be the Cantor-Lebesgue function (see SS Ch.1, Exercise 2) and let g : [0 , 1] → [0 , 2] be the function g ( x ) = x + f ( x ). Prove that g is a homeomorphism and m ( g ( C )) = 1, where C is the Cantor set. Then show that if B is a Lebesgue nonmeasurable subset of g ( C ) (see SS Ch.1, Exercise 32(b)), then g-1 ( B ) is not Borel but is Lebesgue measurable. 1...
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