课程名称︰实分析一 课程性质︰数学所必修 应数所选必 课程教师︰沈俊严 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2017/11/8 考试时限（分钟）：110 mins 试题 : 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. -- 枯藤 老树 昏鸦 枯坐 乾等 涂鸦 小桥 流水 人家 小睡 收拾 回家 古道 西风 瘦马 佛祖 耶稣 阿拉 夕阳西下 断肠人在天涯 考卷发下 重修人在台大 --

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