332 FOR FÄ°NAL

332 FOR FÄ°NAL - T. Liggett Mathematics...

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Unformatted text preview: T. Liggett Mathematics 131C – Final Exam Solutions June 7, 2010 (25) 1. (a) State Fatou’s Lemma. See Royden, page 86. (b) State the Bounded Convergence Theorem. See Royden, page 84. (c) Use Fatou’s Lemma to prove the Bounded Convergence Theorem. Suppose | f n | ≤ M for each n . Then M + f n and M- f n are nonnegative functions. By Fatou, Mm ( E )- Z E f = Z E ( M- f ) ≤ lim inf n →∞ Z E ( M- f n ) = Mm ( E )- lim sup n →∞ Z E f n and Mm ( E )+ Z E f = Z E ( M + f ) ≤ lim inf n →∞ Z E ( M + f n ) = Mm ( E )+lim inf n →∞ Z E f n . Since m ( E ) < ∞ , lim sup n →∞ Z E f n ≤ Z E f ≤ lim inf n →∞ Z E f n , so Z E f = lim n →∞ Z E f n . (20) 2. Evaluate lim n →∞ Z n 1- x n n e ax dx for each real a . Justify each of your steps. The integrand converges to e ( a- 1) x 1 [0 , ∞ ) ( x ) for each x . Since 1- t ≤ e- t for all t , the integrand is dominated by e ( a- 1) x 1 [0 , ∞ ) ( x ), which is integrable for a < 1. By the Dominated Convergence Theorem, lim n →∞ Z n 1- x n n e ax dx = Z ∞ e ( a- 1) x dx = 1 1- a for a < 1. Since Z n 1- x n n e ax dx is increasing in a , it follows that lim n →∞ Z n 1- x n n e ax dx = ∞ for a ≥ 1....
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332 FOR FÄ°NAL - T. Liggett Mathematics...

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