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 ________________ 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 n 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. n 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. n 4. (15%) Suppose A and B are Lebesgue nonmeasurable subset in |R . Prove or disprove A ∪ B is Lebesgue nonmeasurable. --

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