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︰2022年01月05日(三) 考試時限：10:20-12:10，共計110分鐘 試題 : 　　　　　　　　　　　 Real Analysis, Fall 2021 Final Exam DEP. ________________ NAME ____________________ ID NUMBER ___________________ Note. E is always measurable in this exam. 1. (25%) Suppose f(x,y) : (0,1) × (0,1) → |R is a continuous function in x for each fixed y and is a measurable function in y for each fixed x. Show that f is a measurable function. 2. (a.) (10%) State the Lusin's theorem. (b.) (15%) Let f : (0,1) → |R be a finite measurable function. Show that for ε > 0, there exist a continuous function g : (0,1) → |R and a subset of (0,1), E, such that | (0,1) \ E | < ε, and f = g on E. n 3. (20%) Let f_k and f be measurable functions on E ⊂ |R such that |f_k|, 2 |f| ∈ L (E). Suppose f_k → f almost everywhere as k → ∞. Show that 2 2 2 lim ∫ |f_k - f| = 0 if and only if lim ∫ |f_k| = ∫|f| . k→∞ E k→∞ E E 4. Determine whether the following statements are true or false. Give a proof or counterexample for your answer. 　．(a.) (10%) Suppose f_k ∈ L(E) for each k ∈ |N. Then ∫ liminf (f_k) ≦ liminf ∫f_k. E k→∞ k→∞ E 　．(b.) (10%) Let {f_k} be a sequence of measurable functions on a measurable set E and f_k → f almost everywhere. Then, for any ε > 0, there exists a closed set F ⊂ E such that | E \ F | < ε and f_k → f uniformly on F. 　．(c.) (10%) Suppose f ∈ L(E), k ∈ |N. Then, lim ∫ f = 0. 　 k→∞ {f>k} --

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