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# fa - α ∈ A B α = α ∈ A B α#2 Show that for...

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Math 8601: REAL ANALYSIS. Fall 2010 Problems for Final Exam on Saturday, December 18, 4pm–6pm, VinH 1. This Final Exam will be based on the material from the textbook, in the following Sections: 0.5–0.6, 1.1–1.5 ,2.1–2.5, 2.6 (Theorems 2.40, 2.41), 3.1–3.2, 3.4. You will have 2 hours (120 min) to work on 8 problems, 3 of which will be selected from the following list. It is recommended to prepare solutions of take-home problems on separate pages, then you can just enclose them together with in-class problems without rewriting. No books, no notes during this exam. Calculators are permitted, however, for full credit, you need to show step-by-step calculations. #1. Let A be an arbitrary set, and for each α A , let an open ball B α R n be defined. Show that there is a finite or countable subset
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Unformatted text preview: [ α ∈ A B α = [ α ∈ A B α . #2. Show that for arbitrary I := [ a, b ] ⊂ (0 , ∞ ), ± ± ln x-(ln x ) I ± ± I ≤ 1 , where f I := 1 b-a b Z a f ( x ) dx. Hint. It is convenient to write ln x-ln y = x Z y dt t . #3. Let f ( x ) be a function in L 1 ([0 , 1]), such that f (0) ≥ , f (1) ≥ , and f ² x + y 2 ³ ≥ f ( x ) + f ( y ) 2 for all x, y ∈ [0 , 1] . Show that f ( x ) ≥ 0 for all x ∈ [0 , 1]. #4. Give an example of inﬁnite measure ν and ﬁnite measure μ on R 1 , such that ν ± μ , and for each δ > 0 there is an interval I ⊂ R 1 satisfying μ ( I ) < δ and ν ( I ) ≥ 1 . Hint. Try dμ = f dx, dν = g dx with diﬀerent positive f, g . Remark. By Theorem 3.5, such an example is impossible if ν is ﬁnite....
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