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.