[試題] 103上 劉豐哲 實分析一 期中考

[myutwo150 (O51owtuym)]

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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※ 編輯: myutwo150 (61.230.136.175), 11/12/2014 15:44:41
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