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finalsolns

finalsolns - Mathematics 105 Spring 2004 — M Christ Final...

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Unformatted text preview: Mathematics 105, Spring 2004 — M. Christ Final Exam Solutions (selecta) 1 (2e) True or false: If K ⊂ R is a compact set of Lebesgue measure zero, and if f : R → R is a homeomorphism (that is, f is continous and invertible, and f- 1 is also continuous), then | f ( K ) | = 0. Solution. Like all the true/false questions, this is false. Take K to be the Cantor set C ⊂ [0 , 1] which is defined by successively deleting middle thirds of intervals. C = ∩ ∞ n =0 C n where C n is a disjoint union of 2 n (particular) intervals of length 3- n . Consider the piecewise linear function f n which is constant on each interval in the complement of C n , and has constant slope equal to 3 n / 2 n on each of the intervals comprising C n . It can be shown that the sequence f n converges uniformly as n → ∞ to a limiting function f , which is necessarily continuous. Consider the function g ( x ) = f ( x ) + x . Then g is strictly increasing and is continuous, so is a bijection from R to R , and also has a continuous inverse. Since g (0) = 0 and g (1) = 2, g maps [0 , 1] bijectively to [0 , 2]. [0 , 1] \ C is a countable disjoint union of intervals I , the sum of whose lengths equals | [0 , 1] \ C| = 1- 0 = 1. On each of those intervals I , g ( x ) = x plus a constant, so each I is mapped to an interval of length | I | . Their images are disjoint, so | g ([0 , 1] \ C ) | = ∑ I | I | = 1. Since g ( C ) = [0 , 2] \ ( g ([0 , 1] \ C ) ) , it follows that | g ( C ) | = 2- 1 = 1 > 0. (4a) Let ( E, A , μ ) be a measure space and for each j = 1 , 2 , ··· let A j ∈ A . Suppose that ∑ ∞ j =1 μ ( A j ) is finite. Let B be the set of all points x ∈ E which belong to infinitely many of the sets A j . Show that μ ( B ) = 0....
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finalsolns - Mathematics 105 Spring 2004 — M Christ Final...

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