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 ∫