課程名稱︰分析導論優二
課程性質︰數學系大二必修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/05/05
考試時限（分鐘）：50
試題 :
n
1. (20%) (a) Let f: |R → |R be a measurable function. Assume that B is a Borel
-1
set of |R. Show that f (B) is a measurable set.
(b) Assume that f is a function from X onto Y. Let B(Y) be a σ-algebra on Y.
Define a σ-algebra on X, B(X), such that f: (X, B(X)) → (Y, B(Y)) is a
measurable function.
2. (10%) If f(x), x ∈ |R, is continuous at almost every point of an interval
[a, b], show that f is measurable on [a, b].
3. (10%) Let D be a dense subset of |R. Let f be an extended real-valued
function defined on |R. Assume that {x ∈ |R: f(x) > a} is measurable for
each a ∈ D. Then f is measurable.