課程名稱︰實分析一
課程性質︰數學研究所必選修、應用數學科學研究所必選修、數學系選修
課程教師︰劉豐哲
開課學院：理學院
開課系所︰數學系、數學研究所、應用數學科學研究所
考試日期︰2013年11月
考試時限：110分鐘
試題 :
Real Analysis I (Fall 2013)
Mid-term Examination
1. (6%) A real-valued function f defined on a metric space M is called lower
semi-continuous at x in M if for every sequence {x_k} in M converging to x
we have
f(x) ≦ liminf f(x_k)
k→∞
Show that f is lower semi-continuous at x if and only if for every given
ε > 0, there is δ > 0 such that f(y) > f(x) - ε whenever ρ(x,y) < δ.
2. (15%) Let f be a real-valued continuous function on R. Show that f is
Lebesgue integrable on R if and only if for every sequence {I_n} of finite
disjoint open intervals, the system {∫ f(x)dx}_n is summable.
I_n
3. (6%) Suppose that μ measures Ω and that A is a μ-measurable subset of Ω.
Show that for any B contained in Ω,
μ(A) + μ(B) = μ(A∪B) + μ(A∩B)
4. (6%) Let μ be a measure on Ω defined by μ(A) = 0 or 1 according as
A = empty set or A ≠ empty set. Find Σ^μ.
n x n -2x
5. (a) (7%) Evaluate lim ∫(1 + ---) e dx.
n→∞ 0 n
1 x^p 1 ∞ 1
(b) (8%) Show that ∫ -----ln(---) dx = Σ --------- (p > 0).
0 1-x x n=1 (p+j)^2
6. (10%) Let {f_n} be a sequence of measurable functions on (Ω,Σ,μ) such
∞ ∞
that ∫ Σ |f_n|dμ < ∞. Show that Σ f_n(x) converges and finite for
Ω n=1 n=1
∞ ∞ ∞
a.e. x, Σ f_n is integrable and ∫ Σ f_n dμ = Σ ∫f_n dμ
n=1 Ω n=1 n=1 Ω
7. (12%) Let (Ω,Σ,μ) be a measure space and {f_n}_(n in N) contained in
L^1(Ω,Σ,μ). Suppose that for μ-a.e. x in Ω, f_n ≧ f_(n+1) ≧ 0 for
∞ n+1 1
each n in N and lim f_n(x) = 0. Show that Σ (-1) f_n in L(Ω,Σ,μ) and
n→∞ n=1
∞ n+1 ∞ n+1
∫ (Σ(-1) f_n )dμ = Σ (-1) ∫f_n dμ.
Ω n=1 n=1 Ω
8. (10%) Let f be a real-valued function defined on [0,1] ×[0,1]. Suppose that
for each x in [0,1], f(x,．) is Lebesgue integrable on [0,1] and that
∂f/∂x exists and is bouned on [0,1] ×[0,1].
Show that for each x in [0,1], ∂f/∂x (x,．) is λ-measurable on [0,1] and
d 1 ∂f
----∫ -----(x,t) dλ(y).
dx 0 ∂x
9. (8%) Let (Ω,Σ,μ) be a measure space and 1≦ p ＜ q ＜ ∞. Suppose that
{f_n}_(n in N) is a sequence converging to 0 in both L^p(Ω,Σ,μ) and
L^q(Ω,Σ,μ) as n→∞. Show that lim ||f || = 0 for each p < r < q.
n→∞ n r
10. (6%) Let (Ω,Σ,μ) be a measure space and f,g are integrable functions on
Ω. Suppose that ∫ fdμ = ∫ gdμ for each E in Σ. Show that f = g
E E
μ-a.e. on Ω.
11. (6%) Let S be a proper subset of a metric space M and for x in M.
Let f(x) = ρ(x,S). Show that f is continuous on M and that the set
{x in M : ρ(x,S) ≦ δ} is closed for each δ≧ 0.
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