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}