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mt2a - = k A n#2 Let f x,y be a function defined on the...

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Math 8601: REAL ANALYSIS. Fall 2010 Problems for Midterm Exam #2 on Wednesday, November 17. This Midterm will be based on the material for the textbook up to (including) Section 2.3. You will have 50 minutes (10:10 am–11:00 am) to work on 5 problems, 2 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. The remaining 3 problems will be included into the next Homework assignment. No books, no notes during this exam. Calculators are permitted, however, for full credit, you need to show step-by-step calculations. #1 (Borel-Cantelli Lemma). Let ( X, M , μ ) be a measure space, and { A n } be a sequence of sets in M . Show that if X n =1 μ ( A n ) < , then μ lim sup n →∞ A n = 0 , where lim sup n →∞ A n := \ k =1 [ n = k
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Unformatted text preview: = k A n . #2. Let f ( x,y ) be a function defined on the unit square { ≤ x ≤ 1 , ≤ y ≤ 1 } , which is continuous in each variable separately. Show that f is a Borel measurable function of ( x,y ). #3. Let f ( x ) be a right continuous function on R 1 , i.e. lim x → x ,x>x f ( x ) = f ( x ) for all x ∈ R 1 . (a). Show that f is measurable. (b). Show that f is continuous a.e. in R 1 . #4 (Problem 21 on p.59). Let ( X, M ,μ ) be a measure space, and let f,f 1 ,f 2 ,... be non-negative functions functions in L 1 ( μ ), such that f n → f a.e. as n → ∞ . Show that if Z f n dμ → Z f dμ, then Z | f n-f | dμ → . #5 (compare with Problem 31(d) on p.60). Show that for a > 0, ∞ Z e-ax x-1 sin xdx = arctan ( a-1 ) . Hint. Apply Theorem 2.27(b)....
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