课程名称︰分析导论优二
课程性质︰数学系大二必修
课程教师︰王振男
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2015/05/05
考试时限（分钟）：50
试题 :
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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.