中原大學九十二學年度博士班入學招生考試─應用數學系，分析

中原大學九十二學年度博士班入學招生考試

92年6月11日　08:30~10:00　應用數學系	誠實是我們珍視的美德，我們喜愛「拒絕作弊，堅守正直」的你！
科目：分析	(共1頁第1頁)

Part I. Advances Calculus
1.	(10分)	Suppose that is bounded and that is defined by Prove that f is Riemann-Stieltjes integrable with respect to over [-1, 1] if and only if .


2.	Suppose that a<b and that is a convex function;－i.e., for all and all .
	(a)(10分)	Show that, if a<x<z<y<b, then .
	(b)(10分)	Prove that, for each , both the limits
	(c)(10分)	Prove that the set is at most countable.

3.	(10分) Prove that converges if ,\|z\|=1, and .


Part II. Real Analysis
1.	Suppose is a measure space and . Let be given.
	(a)(10分)	Prove that there exists a such that whenever and .
	(b)(10分)	Prove that there exists a measurable set such that.

2.	(10分)	Suppose that and define Show that .

3.	Let be a measure space with .
	(a) (10分)	Prove that, if , then .
	(b) (10分)	Assume that . Show that .

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