ANALYSIS QUALIFYING EXAM PROBLEMS BRIAN LEARY ...

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$Math 3C Homework 3 Solutions$

30 <strong>ANALYSIS</strong> QUALS Problem 6: Define for each n = 1, 2, 3, . . . , the Cantor-like set Cn as [0, 1] with its central open interval of length 1 2n · 1 3 removed, then with the two central open intervals of length 1 2n · 1 32 removed from the remaining two closed intervals and so on (at the jth stage, 2j−1 intervals of length 1 2n · 1 3j are removed), continuing with j = 1, 2, 3, . . .. (a) With µ=Lebesgue measure, show that µ([0, 1]\ ∞ n=1 Cn) = 0. (b) Show that if E is a subset of [0, 1] which is not Lebesgue measurable (you may assume such an E exists without proof), then for some n ≥ 1, E ∪ Cn fails to be Lebesgue measurable. (c) Use part (b) to show that there is a continuous, strictly increasing function f : R → R with f(R) = R and a Lebesgue measurable set A ⊂ R such that f(A) is not Lebesgue measurable. Solution. (a) Cn is constructed by removing intervals of total length 2−n 2 3 at step j. But 1 3 ∞ j=0 j 2 = 3 1 1 − 2 3 = 3, so Cn has a total of 2 −n removed. Thus, Cn has total measure 1−2 −n , so n Cn has measure 1, and hence, µ([0, 1]\ ∞ n=1 Cn) = 0. (b) E not measurable implies that there exists ε > 0 and a set A such that µ(E) < µ(E∩A)+µ(E∩A c )−ε. Choose n such that µ(Cn) > 1−ε/10. Then µ(E ∩ Cn) < µ(E) < µ(E ∩ Cn ∩ A) + µ(E ∩ Cn ∩ A c ) − 4ε/5, so E ∩ Cn is not measurable. (c) Let f be a strictly increasing homeomorphism from C0 to Cn, where n as in part (b). Such a function exists: both sets are homeomorphic to {0, 1} N by considering at each point in a Cantor set, whether it is in the left or right side at each stage of removing central intervals. Thus, just as the strictly monotonic Cantor-Lebesgue function can be constructed for C0 onto [0, 1], so too can such a function be created for Cn, and taking an appropriate composition yields f. Extend f linearly to all of R. Then f −1 (E∩Cn) is a subset of C0, which has measure zero. Therefore f −1 (E ∩ Cn) is measurable. However, by part (b), E ∩ Cn is not measurable. Problem 8: Suppose f : U → C is a holomorphic function with intU |f| 2 < ∞ where U = {z ∈ C : 0 < |z| < 1} and the integral is the usual R 2 -area integral. Prove that f has a removable singularity at z = 0. Solution. Let w be a point in the annulus around z = 0 of radii ε and 1/2, for 0 < ε < 1/2. Then by Cauchy integral formula, we have that f(ξ) f(ξ) f(w) = dξ − ξ − w ξ − w dξ. |ξ|=1/2 |ξ|=ε j−1·
δ 0 Then f(ξ) δf(w) = δ dξ − |ξ|=1/2 ξ − w and thus f(w) − |ξ|=1/2 <strong>ANALYSIS</strong> QUALS 31 f(ξ) ξ − w dξ ≤ 1 δ δ 0 δ 0 |ξ|=ε |ξ|=ε f(ξ) ξ − w dξdε f(ξ) ξ − w dξdε . We claim that this tends to 0 as δ → 0, which would prove that f(w) = f(ξ) 1 |ξ|=1/2 ξ−w dξ for |w| < 2 , and this is holomorphic, so any singularity at 0 would be removable. But using Fubini, Cauchy-Schwarz, and basic estimations, we have that f(ξ) ξ − w dξ = δ 2π f(εe iθ ) εeiθ − w iεeiθ dθdε |ξ|=ε ≤ 0 0 δ 2π 0 ≤ 2 |w| ≤ 2 |w| ≤ 2 |w| 0 2π δ 0 2π 0 |f(εeiθ )| |εeiθ − w| εdθdε 0 δ 0 |f(εe iθ )|εdεdθ |f| {0
Page 1 and 2: ANALYSIS QUALIFYING EXAM PROBLEMS B
Page 3 and 4: ANALYSIS QUALS 3 Proof. (a) First,
Page 5 and 6: ∂∆n ANALYSIS QUALS 5 0, 1, 2, .
Page 7 and 8: ANALYSIS QUALS 7 Thus, as |h ′ (0
Page 9 and 10: ANALYSIS QUALS 9 But by the geometr
Page 11 and 12: Then Γ (j) k ANALYSIS QUALS 11 f(
Page 13 and 14: ANALYSIS QUALS 13 = 1 √ |f(w)|
Page 15 and 16: ANALYSIS QUALS 15 Then B has the s
Page 17 and 18: ∂Q ANALYSIS QUALS 17 Problem 8: L
Page 19 and 20: ANALYSIS QUALS 19 Consider ˆxn ∈
Page 21 and 22: ANALYSIS QUALS 21 b) Suppose S −
Page 23 and 24: ANALYSIS QUALS 23 Problem 12: Let f
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Page 27 and 28: ANALYSIS QUALS 27 Solution. Picard
Page 29: ANALYSIS QUALS 29 Problem 2: Is eve
Page 33 and 34: ANALYSIS QUALS 33 But as D is compa
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Page 37 and 38: Problem 10: Let the power series f(
Page 39 and 40: ANALYSIS QUALS 39 where the inequal
Page 41 and 42: ANALYSIS QUALS 41 Problem 8: Let f
Page 43 and 44: ANALYSIS QUALS 43 Winter 2007 Probl
Page 45 and 46: Then for N sufficiently large, we h
Page 47 and 48: ANALYSIS QUALS 47 see that on ( √
Page 49: ANALYSIS QUALS 49 If z0 ∈ K, choo

ANALYSIS QUALIFYING EXAM PROBLEMS BRIAN LEARY ...

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