100 學年度下學期數學系博士班資格考試 (實變分析)

100 學年度下學期數學系博士班資格考試 (實變分析)

本試題卷共 2 頁，計 10 題計算證明題，每題 10 分，合計 100 分。

1. Prove the Carath ´eodory theorem: A set E is measurable if and only if for every set A,

|A|e=|A ∩ E|e+|A \ E|e.

(Note: |A|e| denotes the outer measure of A.)

2. Prove that the set of points at which a sequence of measuable real-valued functions con- verges (to a finite limit) is measurable.

3. Let f be a function which is upper semi-continuous and finite on a compact set E. Show that if f is bounded above on E. Show also that f assumes its maximum on E, that is, that there exists x₀∈ E such that f (x0)≥ f (x) for all x ∈ E.

4. Let f ∈ L(0,1). Show that x^kf (x)∈ L(0,1) for k = 1,2,..., and ^∫₀¹x^k f (x) dx→ 0 as k→ ∞.

5. Let E be a measurable subset ofR² such that for almost every x∈ R¹, {y | (x,y) ∈ E}

hasR¹-measure zero. Show that E has measure zero, and the for almost every y∈ R¹, {x | (x,y) ∈ E} has measure zero.

6. (a) Write out the definition of the essential supremum∥ f ∥∞of a real-valued measurable function f on a measurable set E.

(b) Let f be a real-valued measurable function on [0, 1]. Prove that lim

p→∞∥ f ∥p=∥ f ∥∞. 7. Let E be a measurable set inRⁿ, and 0 < p < q≤ ∞.

(a) Prove that L^p(E)∩ L^∞(E)⊂ L^q(E).

(b) Prove that if|E| < ∞, then L^q(E)⊂ L^p(E).

8. Let f , g∈ L²(Rⁿ). Prove that f + g∈ L²(Rⁿ) and∥ f + g∥2≤ ∥ f ∥2+∥g∥2.

(背面尚有試題)

1

9. Let{φk} be an orthonormal system in L²[0, 1], and{ck} be the Fourier series of a function f ∈ L²[0, 1] with respect to the system{φk}.

(a) Prove that the Bessel's inequality ( ∞

k=1

∑

|ck|² )_1/2

≤ ∥ f ∥2holds.

(b) Find a necessary and sufficient condition so that the Parseval's identity ( ∞

k=1

∑

|ck|² )1/2

=

∥ f ∥2holds, and prove your answer.

10. Let C[0, 1] denote the set of all real-valued continuous functions on [0, 1], and the linear operator T : C[0, 1]→ R be defined by T( f ) = f (1) for all f ∈ C[0,1].

(a) Prove that T is a continuous linear functional on C[0, 1].

(b) Prove that there exists an extension T^∗: L^∞[0, 1]→ Rⁿof T such that T^∗is a contin- uous linear functional on L^∞[0, 1], but there is no g∈ L¹[0, 1] satisfying

T^∗( f ) =

∫

[0,1]

( f× g)dx for all f ∈ C[0,1].

(試題結束)

2

101 學年度上學期數學系博士班資格考試 (實變分析)

本試題卷共 2 頁，計 10 題計算證明題，每題 10 分，合計 100 分。

1. Let E be a measurable subset ofR, with |E| > 0. Prove that there exists a positive real numberε such that (−ε^,ε⁾⊂ E − E, where

E− E = {x − y | x,y ∈ E}.

2. Prove or disprove:

(a) Any function f : [a, b]→ R of bounded variation is measurable.

(b) Any upper semicontinuous function f : [a, b]→ R is measurable.

3. Let E be a measurable set inRⁿof finite measure. Prove that f : E→ R is measurable if and only if for anyε> 0, there exists a closed subset F of E such that|E \ F| <ε^{, and f} is continuous on F.

4. (a) State without proof the Egorov’s theorem.

(b) Let⟨ fk⟩ be a sequence of measurable functions on a measurable set E with |E| < ∞.

If f_k converges to f a.e. in E, and sup

k

| fk− f | ∈ L(E), prove that lim

k→∞

∫

E

f_k=

∫

E

f. 5. Let f : [0, 1]×[0,1] → R satisfy for each x ∈ [0,1], f (x,y) is a Lebesgue integrable func-

tion of y, and ∂^{f (x, y)}

∂^x is a bounded function of (x, y). Prove that∂^{f (x, y)}

∂^x is a measurable function of y for each x∈ [0,1], and

d dx

∫

[0,1]

f (x, y) dy =

∫

[0,1]

∂^{f (x, y)}

∂^{x dy.}

6. (a) State the definition for a finite function f on a finite interval [a, b] to be absolutely continuous.

(b) Show that the function f (x) = x^α is absolutely continuous on every bounded subin- terval of [0,∞) wheneverα^{> 0.}

7. Let a₁, a₂, . . . , a_N be non-negative real numbers, p₁, p₂, . . . , p_N be positive real numbers with∑^Nj=1(1/p_j) = 1. Show that

∏

N j=1

a_j≤

∑

^N

j=1

aj

pj

.

(背面尚有試題)

1

8. Let ℓ^∞ denote the normed linear space of all bounded real sequences. Is ℓ^∞ separable?

Justify your answer.

9. Suppose that f_k, f ∈ L², and that^∫ f_kg→^∫ f gfor all g∈ L². If∥ fk∥2→ ∥ f ∥2, show that f_k→ f in L²norm.

10. Let Σ be a σ-algebra on a set S , {Ek} be any sequence of sets in Σ, andϕ ^{be a non-} negative additive set function onΣ. Prove that

ϕ^{(lim inf}_k→∞ ^Ek

)≤ liminf

k→∞ ϕ^(Ek).

(試題結束)

2

103 學年度數學系博士班資格考試

(實 變 分 析)

※ 本試題卷共 8 題證明題

1. (a) Prove that every Borel measurable subset inⁿis Lebesgue measurable.

(b) Prove that there is a Lebesgue measurable subset inⁿis not Borel measurable.

(10%)

2. Prove or disprove (Please explain your answer):

(a) If ^{f :}

 

^{a b,  } is a function of bounded variation, then f is Lebesgue measurable.

(b) If E is a Lebesgue measurable subset of  , with E 0, then there exist x y,  E with x y such that x is a rational number. y

(c) If for each rational number a , the set



^{xn f x( )a}



is Lebesgue measurable, then f :ⁿ  is Lebesgue measurable.

(d) There exists a Riemann integrable function f : [0,1][0,1]such that f is continuous

at each rational point and discontinuous at each irrational point of [0,1] . (e) If f is Lebesgue integrable over E , then f is finite a.e. in E . (30%)

3. Prove that if ^{f :}

 

^{a b,  } is a function of bounded variation, then f can be written as f  g  , where g is absolutely continuous and h is singular, which are unique up to h additive constants. (10%)

4. Prove that if f  L E^p( )

and f  , then0 ¹

0

p p

E f  p ^  ^ d

 

^{( ) , where is the}

distribution function of f , defined by^{ ( ) }



^{x  E f x( )  }



. (10%) 5. Prove that if ( )

f  Lp  , where1  p   , then for every  0there is a continuous functiong with compact support such that

f  g p  . (10%) 

6. Prove that if f L(ⁿ), then the definite integral ( ) ( )

F E 



E f x dx is absolutely continuous with respect to Lebesgue measure. (10%)

7. For f g, L(ⁿ), we define the convolution of f and g by (f g x)( )



_n f x( y g y dy) ( ) for xⁿ .

Prove that f  g L R( ⁿ), and f g₁  f ₁ g₁. (10%)

8. Let

 

k be an orthonormal system in ²[ , ]0 1

L , and

 

ck be a sequence in ²( )

 R . Prove that there exists ²[ , ]0 1

f  L such that

1

( )

k k k

c x





 is the Fourier series of f with respect to the orthonormal system

 

k . (10%)