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 +