課程名稱︰實分析一
課程性質︰數學研究所必修課；應用數學所必選課
課程教師︰陳逸昆
開課學院：理學院
開課系所︰數學研究所
考試日期︰2021年11月17日(三)
考試時限：10:20-12:10，共計110分鐘
試題 :
　　　　　　　　　　　　　Real Analysis I, Fall 2021
　　　　　　　　　　　　　　　Midterm Exam
DEP. ___________________ NAME ____________________ ID NUMBER ________________
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1. (a.) (15%) Suppose A and B are disjoint nonempty compact subsets of |R .
Show that
dist(A,B) = inf | x - y | > 0
x∈A, y∈B
(b.) Suppose A and B are disjoint closed nonempty sets. Does the conclusion
in (a.) still hold? Give a proof or a counterexample. (10%)
2. (a.) (15%) Suppose a subset of |R, A, has positive Lebesgue outer measure,
show that A has a Lebesgue nonmeasurable subset. (b.) (15%) Show that there
exists a continuous transformation that maps a Lebesgue measurable set to a
Lebesgue nonmeasurable set.
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3. (a.) (20%) Suppose A⊆E⊆B⊆|R and both A and B are Lebesgue measurable and
|A| = |B|. Then if |B| < ∞, show that E is also measurable. (b.) (10%) Suppose
|B| = ∞. Does the conclusion in (a.) still hold? Give a proof or a
counterexample.
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4. (15%) Suppose A and B are Lebesgue nonmeasurable subset in |R . Prove or
disprove A ∪ B is Lebesgue nonmeasurable.