課程名稱︰實分析一 課程性質︰選修 課程教師︰劉豐哲 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2014/11/12 考試時限（分鐘）：110分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. (15%) Show that {(-1)^(n+m) e^-(n+m)} n,m∈Ｎ is summable and find its sum. 2. (20%) Suppose that f is a continuous function on Ｒ. Show that f is Lebesgue integrable if and only if {∫ f(x) dx} is summable for every disjoint I_n sequence {I_n} of finite open intervals. 3. (15%) For measurable functions f on (Ω,Σ,μ) let N(f) = sup αμ( {|f| > α} ). α>０ Show that N(liminf |f_n|) ≦ liminf N(f_n) n→∞ n→∞ for any sequence {f_n} of measurable functions. 4. (20%) A family {f_α} of integrable functions on a measure space (Ω,Σ,μ) is called uniformly integrable if for any ε>０, there is δ>０ such that if Ａ＜Σ with μ(Ａ)≦δ, then ∫|f_α| dμ≦ε for all α. Show that if μ(Ω) < ∞ and {f_n} is a uniformly integrable sequence of functions on Ω which converges a.e. to an integrable function f on Ω, then lim ∫ |f_n - f| dμ＝０. n→∞ Ω 5. Evaluate the following limits: n (a) (7%) lim ∫ (1 + x/n)^n e^(-2x) dx. n→∞ 0 ∞ sin(x/n) (b) (8%) lim ∫ ----------- dx. n→∞ 0 x^(3/2) 6. (15%) Let μ be an outer measure on Ω. For Ａ＜Ω, define μ_e(Ａ) := inf {μ(Ｂ): Ａ＜Ｂ, Ｂ is μ-measurable}; μ_i(Ａ) := sup {μ(Ｂ): Ａ＞Ｂ, Ｂ is μ-measurable}. Suppose that μ(Ａ) < ∞, show that Ａ is μ-measurable if and only if μ_e(Ａ)＝μ_i(Ａ). 註：集合Ａ包含於Ｂ的符號用全形Ａ＜Ｂ替代 --

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