RealAnalysis2010aug - (b) Z [ n,n +1] | f | 0 as n 0. (c) n...

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Analysis Qualifying Exam August 2010 You must justify your answers in full detail, and explicitly check all the assumptions of any theorem you use. 1. Assume that f,f 1 ,f 2 , ··· ∈ L 1 ( R ) (Lebesgue measure), and that as n → ∞ (i) f n f pointwise on R and (ii) k f n k 1 → k f k 1 . Prove that for any any measurable set E R , lim n →∞ Z E f n = Z E f . 2. Let f L 2 [1 , ) (Lebesgue measure). For each of the following statements, if the statement is true, prove it, while if false give a counterexample. (a) If f is continuous then f ( x ) 0 as x → ∞ . (Do not assume continuity for parts (b),(c) and (d).)
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Unformatted text preview: (b) Z [ n,n +1] | f | 0 as n 0. (c) n Z [ n,n +1] | f | 0 as n 0. (d) lim inf n n Z [ n,n +1] | f | = 0 3. Let f L 2 (0 , ) (Lebesgue measure). Prove the following: (a) Z x f ( t ) dt x 1 / 2 k f k 2 for x > 0. (b) lim x x-1 / 2 Z x f ( t ) dt = 0. 4. Dene f ( x,y ) = ( x-4 / 3 sin( 1 xy ) if 0 < y < x < 1 otherwise Prove or disprove: Z 1 Z 1 f ( x,y ) dxdy = Z 1 Z 1 f ( x,y ) dydx . 1...
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