臺灣大學數學系
八十七學年度第二學期碩博士班資格考試試題 分析
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A. Choose 4 from the following 5 problems
Let and be measurable functions on . Assume
for some . Show that converges in measure to on .
1.
Let be a measurable set in and .
(a) Show that is measurable.
(b) Show that has measure zero if has measure zero.
2.
Let be all of the rational numbers in . Let .
(a) Show that exists and a.e. on . (b) Show that for a.e.
, we have exists and for a.e. .
3.
Let be a measurable function on . Determine which of the following conditions implies that
(1) for all . (2)
for all .
4.
Show that every nonempty, closed, convex set in a Hilbert space contains a unique element of smallest norm.
5.
B. Choose 1 from the following 2 problems
Suppose is an entire function and is a nonnegative integer. Show that if
for some positive constants and , then is a polynomial of degree at most . 1.
Evaluate the intrgral 2.
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