課程名稱︰實分析二 課程性質︰數學研究所必修課；應用數學所研究所必選課 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︰2018年04月25日(三) 考試時限：10:20-12:10，共計110分鐘 試題 : 　　　　　　　　　　REAL ANALYSIS II EXAM 1. 2018/04/25 Do the following problems and write your arguments as detail as possible. 　1. (10%) Let (E,Σ,μ) be a measure. Given two measurable functions f,g such 　that ∫fdμ = ∫gdμ for every A∈Σ. Prove or disprove f = g a.e. A A 1 θ 1-θ 　2. (15%) Let 1＜p＜r＜q＜∞ and define θ by --- = ---- + ------. Show that 　 θ 1-θ r p r 　　|| f || ≦|| f || || f ||　. 　　　　　 r　　　　 p　　　 q 　　　　　　　　　 1 1 1 　3. (10%) Assume --- = --- + --- - 1 for some 1≦p,q≦∞. Show that 　　　　　　　　　 r p q 　　 || f＊g || ≦|| f || || f || . r　　　　 p　　　　q 　　　　　　　　　　　p 　4. (15%) Suppose f∈L(|R) and is continuous at x . Show that its Poisson 　　　　　　　　　　　　　　　　　　　　　　　　 0　　　 　integral f(x,y)→f(x ) when (x,y)→x with y>0. (recall: Poisson kernel is 　　　　　　　　　　　0 0 １ １ ---- ------) π 1+x^2 　5. (15%) Given a measure space (E,Σ,μ) with μ(E) > 0 and a bounded 　non-constant measurable function f(x). Show that there exists λ∈|R such 　that μ({f≦λ})>0 and μ({f＞λ})>0. 　6. (15%) Let (E,Σ,μ) be a measure space with μ(E) < ∞. Suppose a sequence 　of measurable functions {fk} converges to f a.e. Show that given ε>0, there 　exists a measurable set A such that μ(E - A) < ε and fk→f uniformly on A. 　7. (10%) Suppose (E,Σ,μ) is a measure space with μ(E)=1. Suppose a 　sequence of measurable functions {fn} and a sequence of numbers {λn} satisfy 　Σλn and Σμ({fn≠λn}) both converge. Prove or disprove Σfn converges a.e. 　8. (10%) Suppose f∈L(|R). Prove that lim ∫f(x)cos(nx)dx = 0. 　　　　　　　　　　　　　　　　　　　n→∞ --

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