臺灣大學數學系
九十五學年度博士班入學考試題 分析
Jun, 2006
1. Let {fn} be a sequence of measurable functions. Show that the set of those x such that {fn(x)} converges is a measurable set.
2. Show that, if fn→ f in measure and if there is an integrable function g such that |fn| ≤ g for all n, thenR |fn− f | → 0.
3. Let f (x) be a L1 function on (−∞, ∞).
(a) Show that lims→0R∞
−∞|f (x + s) − f (x)|dx = 0.
(b) Is it true that lims→0R∞
−∞|f (x + sx) − f (x)|dx = 0?
4. Let f be a L1function on (−∞, ∞) and g(x) =R∞
−∞exp {−(y − x)2}f (y)dy.
(a) Show that g(x) is differentiable.
(b) Show that g(x) ∈ Lp for all p ≥ 1.
(c) Are the functionsR∞
−∞exp {−(y − x)2}f (y + x2)dy andR∞
−∞exp {−(y − x)2}f (yx)dy differentiable in x?
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