臺灣大學數學系

(1)

臺灣大學數學系

的學年度下學期博士班資格考試題

c 科目:實分析 201 1.02.24

1. (10 points)

Prove the following in七egral version of Minkowski's inequality for 1 三 p< ∞:

[J I J f(x， y)伽I

^P

dy]l/p 三 J[Jlf叫

2. (10 points)

Prove that L∞ (E) is not separable for any measurable setE with positive measure.

3. (10 points)

Ptove that a continuous function defined on a closed interval is uniformly continuous.

4:(10 points)

Prove that a function defined on a closed interval is Riemann integrable.

5. (30 points)

(i) State and prove the simple Vitali covering lemma.

(ii) Prove that if

f

^{ε L(lR}ⁿ⁾and

f

^hωcompactsupport, then the set

Af﹒ (α) = {x ε lRnlf﹒ (x) > α}

is bounded in lRⁿ, where

r

is the Hardy-Littlewood maximum function of

f.

(iii) Under the 品sumption of (ii)

,

show th此 there exists a positive constant c ind令

. pendent of

f

and

αsuch thatωf﹒ (α)

=

IAf﹒ (α)1 三;三 JR"lfl

6.τ"'rue or False. For each of 七he following statements, prove it if the statement 泊is 七rue;

O her give a counter example. (30 points)

呦七出川se ，

的 Suppose f(x) = 玄立。 αjX

^j

，

where

αj 巴 R1 ， j

=0, 1,2,

... i mdZ立。|句 1=

10252010. Then

f(x)

is of bounded variation on the closed interval

[0，可.

b) Suppose that g(x) is a bounded function on [0,1] suc4 that for any positive i

<

1,

V[g; ε ， 1]

<

M. Then V[g; 0， 1] 三 M， where V[g; α，叫 is the variation of 9 over [a ,b].

C吋) Su叩1中仰記出前吋pposethat f 叫)恤d <þ圳(x 缸.ree bou只(x 叫) a昀 ∞叮叫吋 I unlded fu肌

1 and 后~

J f(仰x)河呦州￠剃仰(伊叫) = d Z

1. Then J~l

f(x) 你 (x)

^{= 2}

d)

fn(x)

•

f (x)

uniformly in lR

=令人 (x) → f(x)

in L1(lR).

e)

fn • f

^{in L}¹^([0,1])

=今 fn → f

pointwise a.e on [0,1].

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