課程名稱︰實分析一 課程性質︰數學研究所必修課；應用數學研究所必選課 課程教師︰沈俊嚴 開課學院：理學院 開課系所︰數學研究所 考試日期︰2020年11月11日(三) 考試時限：10:20-12:10，計110分鐘。 試題 : 　　　　　　　　　REAL ANALYSIS EXAM 1. 2020/11/11 Do the following problems and write your arguments as detail as possible. 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 n 1. (15%) Assume f is a complex-valued measurable function defined on |R . Define the essential range R_f of f to be the set of all z∈C such that 　 {x; |f(x)-z| <ε} has positive measure for all ε> 0. 　 (a) Prove that R_f is closed. 　 (b) Let M be the infimum number such that |f(x)|≦M for almost every x. 　　　 Assume M <∞, show that R_f is compact and M = max{|z|; z∈R_f}. 2. (15%) Let f_n(x) = exp(-n|1-sin(x)|) defined on (0,∞). Prove or disprove 　 f_n converges in meausre to 0. 3. (15%) Suppose {A_n} is a sequence of disjoint measurable sets of [0,1] with 　 ∪A_n = [0,1]. If {B_n} is a sequence of measurable subsets of [0,1] such that lim |B_n ∩ A_n| = 0 for all k. Prove that lim |B_n| = 0. n->∞ n->∞ 4. (15%) Let g(x) be a bounded measurable function with the property that lim ∫g(nx)dx = 0, n->∞ E for any measurable set E with finite measure. If f is Lebesgue integrable on |R, does lim ∫ f(x)g(nx)dx = 0? n->∞ |R Prove or disprove the result. 5. (15%) Suppose M is an σ-algebra with infinite sets. Can M contain only countable many sets? Prove or disprove the result. 6. (15%) Suppose {f_n} is a sequence of measurable functions on [0,1] and each f_n is finite a.e. Prove that we can find a sequnece of positive numbers c_n such that lim c_n f_n(x) = 0 a.e. on [0,1]. n->∞ 7. (10%) The two statements below, which one implies the other? (a) f is continuous a.e. on [0,1]. (b) There is a continuous function g on [0,1] such that g = f a.e. --

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