課程名稱︰實分析一
課程性質︰數學所必修 應數所選必
課程教師︰沈俊嚴
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/11/8
考試時限（分鐘）：110 mins
試題 :
Do the following problems and write your arguments as detail as possible.
1.(10%)
(a) Suppose E_1 and E_2 are nonmeasurable sets in R^n. Prove or disprove
E_1 ∪E_2 is nonmeasurable.
(b) Prove or disprove the following statements.
B is not Lebesgue measurable if and only if there exists ε> 0 such that
for every Lebesgue measurable set A ⊂B, |B-A|_e ≧ε.
2.(15%)
Let f_n(x) = e^{ -n|1-sinx| } defined on (0,∞). Prove or disprove f_n converge
in measure to 0.
3.(15%)
In class we already showed that if f is a bounded and monotone increasing
function on R^1, then the set of its discontinuities is countable. Now if f
is only assumed to be monotone increasing, do we have the same conclusion ?
Prove or disprove your result.
4.(10%)
Show that there exists a set in R^1 which is Lebesgue measurable but not
Borel measurable.
5.(15%)
Let E ⊂R^1 with |E| > 0. Show that the set D = { x-y : x,y ∈E } contains
an interval centered at 0.
6.(10%)
Let X be an uncountable set, and M be the collection of sets E ⊂X such that
either E is countable or E^c is countable. Prove or disprove M is a σ-algebra.
7.(10%)
Suppose f is a function defined on R^1 with the property that the set f^{-1}(c)
is Lebesgue measurable for every c ∈R^1. Prove or disprove f is measurable.
8.(15%)
Suppose {f_k} is a sequence of measurable functions on a measurable set E.
Assume that f is a function on E, and let E_k = { x ∈E : f(x)-f_k(x) > 1/k^2 }.
Suppose |E_k|_e < 1/k^2, ∀k. Prove or disprove f is a measurable function.