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| 实变函数与泛函分析基础第三版(程其襄) 课后答案 |
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????????????????????????????????????????
1. A∪(B∩C) = (A∪B)∩(A∪C).
x∈ (A∪(B∪C)). x∈A, x∈A∪B,x∈A∪C, x∈ (A∪B)∩(A∪C).
x∈B∩C, x∈A∪B x∈A∪C, x∈ (A∪B)∩(A∪C),
A∪(B∩C)? (A∪B)∩(A∪C).
x ∈ (A∪B)∩ (A∪C). x ∈ A, x ∈ A∪ (B∩C). x A,
x∈ A∪B x∈ A∪C, x∈ B x∈ C, x∈ B∩C, x∈ A∪ (B∩C),
(A∪B)∩(A∪C)?A∪(B∩C). A∪(B∩C) = (A∪B)∩(A∪C).
2.
(1)A?B =A?(A∩B) = (A∪B)?B;
(2)A∩(B?C) = (A∩B)?(A∩C);
(3)(A?B)?C =A?(B∪C);
(4)A?(B?C) = (A?B)∪(A∩C);
(5)(A?B)∩(C?D) = (A∩C)?(B∪D);
(6)A?(A?B) =A∩B.
(1)A?(A∩B) =A∩ (A∩B) =A∩( A∪ B) = (A∩ A)∪(A∩ B) =A?B;
s s s s s
(A∪B)?B = (A∪B)∩ B = (A∩ B)∪(B∩ B) =A?B;
s s s
(2)(A∩B)?(A∩C) = (A∩B)∩ (A∩C) = (A∩B)∩( A∪ C) = (A∩B∩ A)∪(A∩
s s s s
B∩ C) =A∩(B∩ C) =A∩(B?C);
s s
(3)(A?B)?C = (A∩ B)∩ C =A∩ (B∪C) =A?(B∪C);
s s s
(4)A?(B?C) =A?(B∩ C) =A∩ (B∩ C) =A∩( B∪C) = (A∩ B)∪(A∩C) =
s s s s s
(A?B)∪(A∩C);
(5)(A?B)∩(C?D) = (A∩ B)∩(C∩ D) = (A∩C)∩ (B∪D) = (A∩C)?(B∪D);
s s s
(6)A?(A?B) =A∩ (A∩ B) =A∩( A∪B) =A∩B.
s s s
3. (A∪B)?C = (A?C)∪(B?C); A?(B∪C) = (A?B)∩(A?C).
(A∪B)?C = (A∪B)∩ C = (A∩ C)∪(B∩ C) = (A?C)∪(B?C);
s s s
(A?B)∩(A?C) = (A∩ B)∩(A∩ C) =A∩ B∩ C =A∩ (B∪C) =A?(B∪C).
s s s s s
∞ ∞
4. ( A ) = A .
s i s i
i=1 i=1
∞ ∞
x∈ ( A ), x∈S, x A , i,x A , x∈ A ,
s i i i s i
i=1 i=1
1????
??????????
∞ ∞ ∞
x∈ A . x∈ A , i,x∈ A , x∈ S,x A , x∈ S, x A ,
s i s i s i i i
i=1 i=1 i=1
∞ ∞ ∞
x∈ ( A ). ( A ) = A .
s i s i s i
i=1 i=1 i=1
5. (1) ( A )?B = (A ?B); (2)( A )?B = (A ?B).
α α α α
α∈Λ α∈Λ α∈Λ α∈Λ
(1) A ?B = ( A )∩ B = (A ∩ B) = (A ?B);
α α s α s α
α∈Λ α∈Λ α∈Λ α∈Λ
(2) A ?B = ( A )∩ B = (A ∩ B) = (A ?B).
α α s α s α
α∈Λ α∈Λ α∈Λ α∈Λ
n?1
6. {A } B =A ,B =A ?( A ),n> 1. {B }
n 1 1 n n ν n
ν=1
n n
A = B ,1≤n≤∞.
ν ν
ν=1 ν=1
i =j, ii i
j?1
B ∩B ?A ∩(A ? A ) =A ∩A ∩ A ∩ A ∩ ∩ A ∩ ∩ A =?.
i j i j n i j s 1 s 2 s i s j?1
n=1
n n
B ?A (1 =i =n) B ? A .
i i i i
i=1 i=1
n n
x∈ A , x∈A , x∈B ? B . x A , i x∈A ,
i 1 1 i 1 n i
n
i=1 i=1
i ?1 i ?1
n n n n n
x A x∈A . x∈A ? A =B ? B . A = B .
i in in i in i i i
i=1 i=1 i=1 i=1 i=1
1
7. A = 0, ,A = (0,n),n = 1,2, , {A }
2n?1 2n n
n
lim A = (0,∞);
n
n→∞
x∈ (0,∞), N, xN 02n
N x A , x∈ lim A , lim A ? (0,∞),
n n n
n→∞ n→∞
lim A = (0,∞).
n
n→∞
lim A =?;
n
n→∞
x ∈ lim A = ?, N, n > N, x ∈ A . 2n? 1 > N
n n
n→∞
1
x∈A , 02n?1 n
n
n→∞
∞ ∞
8. lim A = A .
n m
n→∞ n=1m=n
∞ ∞ ∞
x∈ lim A , N, n>N,x∈A , x∈ A ? A ,
n n m m
n→∞ m=n+1 n=1m=n
∞ ∞ ∞ ∞ ∞
lim A ? A . x ∈ A , n, x ∈ A , m≥ n,
n m m m
n→∞ n=1m=n n=1m=n m=n
x∈A , x∈ lim A .
n n
n→∞
∞ ∞
lim A = A .
n m
n→∞
n=1m=n
2
△△?
?△△△△????△△△??????
9. (?1,1) (?∞,+∞)
π
? : (?1,1) → (?∞,+∞). x ∈ (?1,1),?(x) = tan x. ? (?1,1)
2
(?∞,∞)
10.
1 1
2 2 2 2
S :x +y + (z? ) = ( ) (0,0,1) xOy M
2 2
(x,y,z)∈S\(0,0,1),
x y
?(x,y,z) = , ∈M.
1?z 1?z
? S M
11. A A
G = { | }, r ,
z z z z
r
z z z
G G
12.
∞
A n n = 1,2, , A = A .A n+1
n n n
n=0
n n +1 0
§4
6,A =a, §4 4,A =a.
n
13. A ( )
A
A : (x,y,r). (x,y) r
x,y r 0 A =a.
14.
f (?∞,∞) E,
(1) x ∈ (?∞,∞), lim f(x + x) = f(x + 0) lim f(x + x) = f(x? 0)
+ ?
x→0 x→0
(2)x∈E f(x+0)>f(x?0).
(3) x ,x ∈E, x 1 2 1 2 1 1 2 2
x∈E, (f(x?0),f(x+0)), (3) E x
11
15. (0,1) [0,1]
3′?′′?????′?????′′′??′?
(0,1) R ={r ,r , } ,
1 2
?
?
??(0) =r ,
1
?
?
?
?
?
?(1) =r ,
2
?
?
?(r ) =r ,n = 1,2,
n n+2
?
?
?
?
?
?(x) =x, x∈ ((0,1)\R),
? [0,1] (0,1)
16. A A
A = {x ,x , } ,A A.A ={x ,x , ,x },A
1 2 n 1 2 n n
∞
n
A . A 2 A = A , A A
n n n
n=1
A
17. [0,1] c.
[0,1] A,[0,1] {r ,r , } ,
1 2
√ √ √
2 2 2
B = , , , , ?A
2 3 n
√ √
2 2
?( ) = , n = 1,2,
2n n+1
√
2
?( ) =r , n = 1,2,
n
2n+1
?(x) =x, x B.
? A [0,1] [0,1] c, A c.
18. A A ={a },
x x x
1 2 3
x c A c.
i
x ∈A ,A =c,i = 1,2, . A R ? . ? A
i i i i i
E a ∈A.?(a ) = (? (x ),? (x ),? (x ), ). ?
∞ x x x x x x 1 1 2 2 3 3
1 2 3 1 2 3
?(a ) =?(a ), i,? (x ) =? (x ). ? x =
x x x x x x i i i i i
1 2 3 i
1 2 3
x , a =a . (a ,a ,a , )∈E ,a ∈R,i = 1,2, , ?
x x x 1 2 3 ∞ i i
i 1 2 3 x x x
1 2 3
x ∈ A , ? (x ) = a a ∈ A, ?(a ) = (? (x ),? (x ), ) =
i i i i i. x x x x x 1 1 2 2
1 2 3 1 2
(a ,a , ), ? A E c.
1 2 ∞
∞
19. A c, n , A c.
n 0 n
0
n=1
∞
E =c, A =E . A ∞ n ∞ n i
n=1
E R x = (x ,x , ,x , )∈E , P (x) =x .
∞ 1 2 n ∞ i i
?
A =P (A ),i = 1,2, ,
i i
i
4???????
? ?
A i i i 2 n ∞
i i
∞ ∞
?
ξ A . ξ ∈ A , i, ∈ A , ξ =P (ξ)∈ P (A ) =A ,
n n i i i i i
i
n=1 n=1
∞
?
ξ ∈ R\A ξ A = E , ξ ∈ E i ,
n ∞ ∞ 0
i
n=1
A =c.
i
0
20. 0 1 T, T c.
T ={{ξ ,ξ , }| ξ = 0or1,i = 1,2, } .
1 2 i
T E ? :{ξ ,ξ , } →{ ξ ,ξ , } , ? T E ?(T)
∞ 1 2 2 3 ∞
A ≤ E = c, (0,1] 2
∞
x∈ (0,1] x = 0.ξ ξ , ξ 0 1, f(x) ={ξ ,ξ , } , f
1 2 i 1 2
(0,1] T f((0,1]) T ≥ (0,1] =c. A =c.
5???′′′′′?′?′′??′?′′′′′′′′
o
ˉ
E E E
1. P ∈ E P U(P,δ) ( P )
0 0 0
o
P P E ( P ), P ∈E
0 1 1 0
P U(P,δ)( P ) U(P,δ)?E.
0 0
P ∈E , P U(P,δ), P U(P )?U(P,δ),
0 0 0 0
P ∈E∩U(P )? E∩U(P,δ) P = P, P P
1 0 1 0 1
P E.
0
P P P E, P U(P )
0 0 1 0 0
P P E, P ∈E .
0 1 0
o
P ∈E , U(P )? E.
0 0
o
P ∈U(P,δ)? E, U(P )? U(P,δ)?E, P ∈E .
0 0 0
1 o
ˉ
2. E [0,1] E R E ,E ,E .
1 1 1
1 1
o
ˉ
E = [0,1], E =?, E = [0,1].
1
1 1
2 2 2 o
ˉ
3. E ={(x,y)|x +y < 1}. E R E ,E ,E .
2 2 2
2 2
2 2 o 2 2 2 2
ˉ
E ={(x,y)|x +y ≤ 1}, E ={(x,y)|x +y < 1}, E ={(x,y)|x +y ≤ 1}.
1
2 1
4. E
3
?
1
?
sin , x = 0,
x
y =
?
0, x = 0
2 o
R E E E .
3
3 3
o
E = E ∪{(0,y)|?1≤y≤ 1}, E =?.
3
3 3
2 o
ˉ
5. R 2 E ,E ,E
1
1 1
o
ˉ
E ={(x,0)|0≤ x≤ 1,},E =?,E = E .
1
1 1 1
ˉ
6. F F =F.
ˉ ˉ ˉ
F F ? F, F = F ∪F =F. F = F, F ?F ∪F = F = F,
F
7.
G F G F G?F = G∩ F
F ?G = F ∩ G
8. f(x) (?∞,∞) a,E ={x|f(x) > a}
E ={x|f(x)≥ a}
1??????????????
?′??????
x ∈ E, f(x ) > a. f(x) δ > 0, x ∈
0 0
(?∞,∞),|x?x |< δ f(x)> a, x∈U(x ,δ) x∈E, U(x ,δ)? E,E
0 0 0
x ∈ E, x → x (n→∞). f(x )≥ a, f(x) f(x ) = lim f(x )≥ a,
n n 0 n 0 n
n→∞
x ∈E, E
0
9.
1 1
F G = x|d(x,F) < ,G x ∈ G ,d(x ,F)< ,
n n 0 n 0
n n
1 1 1
y ∈ F, d(x ,y ) = δ < . ( y∈ F,d(x ,y)≥ , d(x ,F) = inf d(x ,y)≥ ,
0 0 0 0 0 0
n n n
y∈F
1
d(x ,F)< ).
0
n
1
= ?δ > 0, x∈U(x , ),d(x ,x) <.
0 0
n
1 1
d(x,y )≤ d(x ,x)+d(x ,y ) < +δ = + ? = .
0 0 0 0
n n
1
d(x,F) = inf d(x,y)≤ d(x,y )< , x∈G . U(x , )? G , G
0 n 0 n n
n
y∈F
∞
1
x∈ G , n,x∈ G ,d(x,F) < . n→∞, d(x,F) = 0. F
n n
n
n=1
∞
x∈ F( x F, y ∈ F, d(x,y )→ 0, x∈ F ? F, ), G ? F.
n n n
n=1
∞ ∞
G ?F,n = 1,2, , G ? F, G = F,F
n n n
n=1 n=1
∞
G G G , G = G ,
n n
n=1
∞ ∞
G = ( G) = ( G ) = G ,
n n
n=1 n=1
G G
n
10. [0,1] 7
[0,1] 7 (0.7,0.8).
[0,1] 7
(0.07,0.08) (0.17,0.18) (0.97,0.98).
[0,1] n 7
(0.a a a 7,0.a a a 8),
1 2 n?1 1 2 n?1
a (i = 1,2, ,n?1) 0 9 7 {a ,a , ,a }
i 1 2 n?1
n?1
∞
A
n
n=1
2????????????????????????
[0,1] 7
∞
A ∪(?∞,0)∪(1,∞) .
n
n=1
A ,(?∞,0),(1,∞) [0,1] 7
n
11. f(x) [a,b] c, E ={x|f(x)≥ c}
E ={x|f(x)≤c}
1
f(x) [a,b] 8 E E E E
1 1
x ∈ [a,b]. f(x) x > 0,x → x ,f(x )≥ f(x ) +
0 0 0 n 0 n 0 0
f(x )≤f(x )? , c =f(x )+, x ∈ E ={x|f(x)≥ c},
n 0 0 0 n
x E( f(x )0 0 0 0
n
12. §2 5: E =?,E =R , E ( ?E =?).
P = (x ,x , ,x ) ∈ E,P = (y , ,y ) E. P = (ty + (1?t)x ,ty +
0 1 2 n 1 1 n t 1 1 2
(1?t)x , ,ty +(1?t)x ),0≤t≤ 1.t = sup{t|P ∈E}. P ∈?E.
2 n n 0 t t
0
P ∈ E. t = 1. t ∈ [0,1] t < t ≤ 1, P E. t ,1 > t >
t 0 0 t n n
0
t ,t →t P ∈ E, P →P , P ∈?E.
0 n 0 t t t t
n n 0 0
P E, t = 0, t ,0 < t < t ,t → t ,P → P ,P ∈ E,
t 0 n n 0 n 0 t t t
0 n 0 n
P ∈?E. ?E =?.
t
0
13. P 1,
P c.
P ( P),
1 2
, = (0.1,0.2),
3 3
1 2
, = (0.01,0.02),
9 9
7 8
, = (0.21,0.22),
9 9
(n)
n?1 n?1
n 2 I ,k = 1,2, ,2
k
(n)
I = (0.a a a 1,0.a a a 2),
1 2 n?1 1 2 n?1
k
a ,a , ,a 0 2. [0,1]?P
1 2 n?1
1, P 1, x∈P, x
a a a
1 2 n
x = + + + + ,
2 n
3 3 3
a 0 2. A, A? P. A? [0,1], [0,1]?P
n
a 1, [0,1]?P A
i
A? P.
3?
A B φ:
∞ ∞
a 1 a
n n
φ : x = → ,
n n
3 2 2
n=1 n=1
ˉ
ˉ
a = 0 2, φ A B 1-1 A c, A? P, P ≥ c,
n
ˉ ˉ
ˉ ˉ
P ≤ c , P = c.
4??△△??
?△△△??
???△△
△△?△△
?
1. E m E < +∞.
? ?
E I E?I. m E≤m I < +∞.
2.
E = {x | i = 1,2, } . > 0, I , x ∈ I , |I | = (
i i i i i i
2
∞ ∞
p S P
p
p
R x I ), I ? E, |I | = .
i i i i i
2
i=1 i=1
?
m E = 0.
? ?
3. E m E > 0, m E c, E
?
E , m E =c.
1 1
?
a = inf x,b = supx, E ? [a,b]. E = [a,x]∩E,a≤x≤ b,f(x) =m E
x x
x∈E
x∈E
[a,b] x> 0
? ?
|f(x+ x)?f(x)| =|m E ?m E |
x+ x x
?
≤|m (E ?E)|
x+
?
≤m (x,x+ x] = x.
x→ 0 f(x+ x)→f(x), f(x) x> 0, x→ 0
f(x? x)→f(x), f(x) [a,b]
? ?
f(a) =m E =m (E∩{a}) = 0
a
? ?
f(b) =m (E∩[a,b]) =m E.
? ? ?
c,c < m E, x ∈ [a,b] f(x ) = c. m E = m ([a,x ]∩E) = c.
0 0 x0 0
?
E =E∩[a,x ]?E. m E =c.
1 0 1
4. S ,S , ,S ,E ?S ,i = 1,2. ,n,
1 2 n i i
? ? ? ?
m (E ∪E ∪ ∪ E ) =m E +m E + +m E .
1 2 n 1 2 n
S ,S , ,S §2 3 1,
1 2 n
n n n n
S P S S
? ?
T, m (T ∩ S ) = m (T ∩S ). T = E , T ∩S = ( E )∩S =
i i i i j i
i=1 i=1 i=1 j=1
n n n n n n
S S S S P P
? ? ? ?
E ,T ∩( S ) = E , m ( E ) =m (T ∩( S )) = m (T ∩S ) = m E .
i i i i i i i
i=1 i=1 i=1 i=1 i=1 i=1
?
5. m E = 0, E
? ? ?
T,T = (E∩T)∪(T ∩ E), m T ≤m (E∩T)+m (T ∩ E).
? ? ? ?
E ∩ T ? E, m (E ∩ T) ≤ m E = 0.T ∩ E ? T,m (T ∩ E) ≤ m T,
? ? ?
m (E∩T)+m (T ∩ E)≤m T.
1??
??????????????????????
??????
? ? ?
m T =m (T ∩E)+m (T ∩ E), E
6. (Cantor)
P [0,1]
1 2
, , , n
3 9
∞
P
n?1 n?1
2 2
, . P [0,1] = 1( ).
n n
3 3
n=1
m[0,1] =m(P ∪([0,1]?P)) =mP +m([0,1]?P).
mP =m[0,1]?m([0,1]?P) = 1?1 = 0, 0.
p ? ? ? ?
7. A,B?R m B < +∞. A m (A∪B) =mA+m B?m (A∩B).
A
? ? ? ?
m (A∪B) =m ((A∪B)∩A)+m ((A∪B)∩ A) =mA+m (B?A).
? ? ?
m B =m (B∩A)+m (B∩ A),
? ? ? ? ?
m B < +∞, m (B∩ A)< +∞, m (B?A) =m B?m (A∩B),
? ? ?
m (A∪B) =mA+m B?m (A∩B).
8. E > 0, G F, F ? E ? G,
m(G?E)<,m (E?F)<.
∞
S
mE <∞ > 0, {I},i = 1,2, , I ?E,
i i
i=1
∞ ∞ ∞ ∞
P S P P
| I |< mE +. G = I , G G ? E, mE ≤ mG≤ mI = | I |<
i i i i
i=1 i=1 i=1 i=1
mE +, mG?mE <, m(G?E)<.
∞
S
mE =∞ E E = E (mE <∞),
n n
n=1
∞
S
E G , G ?E m(G ?E )< . G = G ,G
n n n n n n n n
2
i=1
∞ ∞ ∞
S S S
G?E, G?E = G ? E ? (G ?E ).
n n n n
n=1 n=1 n=1
∞
[
m(G?E)≤ m(G ?E )<.
n n
n=1
E E > 0 G,G? E, m(G? E)<.
G? E =G∩E =E∩ ( G) =E? G,
F = G, F m(E?F) =m(G? E)<.
q
9. E?R , {A },{B }, A ?E?B m(B ?A )→ 0(n→∞),
n n n n n n
E
2
???
??
∞ ∞
T T
i, B ? B , B ?E ? B ?E. E ?A ,B ?E ? B ?A ,
n i n i i i i i
n=1 n=1
i,
!
∞
\
? ? ?
m B ?E ≤m (B ?E)≤m (B ?A ) =m(B ?A ).
n i i i i i
n=1
∞ ∞
T T
?
i→∞, m(B ?A )→ 0, m B ?E = 0. B ?E B
i i n n n
n=1 n=1
∞ ∞ ∞
T T T
B E = B ? B ?E
n n n
n=1 n=1 n=1
p
10. A,B?R ,
? ? ? ?
m (A∪B)+m (A∩B)≤m A+m B.
? ? ? ?
m A = +∞ m B = +∞, m A< +∞ m B < +∞
? ?
G G G , G ?A,G ?B, mG =m A,mG =m B.
δ 1 2 1 2 1 2
? ?
m (A∪B)≤m(G ∪G ),m (A∩B)≤m(G ∩G ).
1 2 1 2
? ? ? ?
m (A∪B)+m (A∩B)≤m(G ∪G )+m(G ∩G ) =mG +mG =m A+m B.
1 2 1 2 1 2
p ?
11. E ? R . > 0, F ? E, m (E?F) < , E
∞
S
? 1
n, F ? E, m (E?F ) < . F = F ,
n n n
n
n=1
F F ?E. n,
1
? ?
m (E?F)≤m (E?F )< .
n
n
?
m (E?F) = 0, E?F
E =F ∪(E?F)
12.
M, ?M, ≤M. c
c,
≥M, =M.
3???
???
?
1. f(x) E r, E[f > r]
E[f =r] f(x)
r,E[f >r] α, {r } α
n
∞
S
E[f > a] = E[f > r ], E[f > r ] E[f > α] f(x) E
n n
n=1
r,E[f =r] f(x) E = (?∞,∞),z (?∞,∞)
√ √
x∈ z,f(x) = 3;x z,f(x) = 2, r,E[f = r] =?
√
E[f > 2] = z f
2. f(x),f (x)(n = 1,2, ) [a,b] k
n
∞
\
1
lim E |f ?f |<
n
k
n→∞
k=1
E f (x) f(x)
n
A E f x ∈ A, k, N, n > N
n
1
|f (x)?f(x)|< ,
n
k
1
x∈ lim E |f ?f |< .
n
k
n→∞
k
∞
\
1
x∈ lim E |f ?f |< .
n
k
n→∞
k=1
h i
∞
T
1 1 1
x∈ lim E |f ?f |< , > 0, k , <, x∈ lim E |f ?f |<
n 0 n
k k k
0 0
n→∞ n→∞
k=1
h i
1 1
N, n > N x ∈ E |f ?f |< , | f (x) ? f(x) |< < ,
n n
k k
0 0
lim f (x) = f(x), x∈A.
n
n→∞
∞
\
1
A = lim E |f ?f |< .
n
k
n→∞
k=1
3. {f } E
n
§1 6, lim f (x) lim f (x) E E[ lim f = +∞]
n n n
n→∞
n→∞ n→∞
f +∞ E[ lim f =?∞] f ?∞
n n n
n→∞
E[ lim f > lim f ] f f (x) E
n n n n
n→∞
n→∞
E?F[ lim f = +∞]?E[ lim f =?∞]?E[ lim f > limf ].
n n n n
n→∞ n→∞
n→∞
1?????
???
E[ lim f = +∞]∪E[ lim f =?∞]∪E[ lim f > lim f ]
n n n n
n→∞ n→∞
n→∞ n→∞
4. E [0,1]
(
x, x∈ E,
f(x) =
?x, x∈ [0,1]?E.
f(x) [0,1] |f(x)|
f(x) 0∈E, E[f ≥ 0] = E 0 E, E[f > 0] = E
f(x)
x∈ [0,1] |f(x)|= x |f(x)| [0,1]
5. f (x)(n = 1,2, ) E a.e. |f |a.e. f.
n n
> 0 c E ?E,m(E\E ) <, E n |f (x)|≤c.
0 0 0 n
mE <∞.
E[| f |= ∞],E[f f] n = 0,1,2, . E = E[f
n n 1 n
∞
S
f]∪( E[|f |=∞]), mE = 0. E?E f (x) f(x). E =E?E ,
n 1 1 n 2 1
n=0
x∈E ,sup| f (x)|<∞.
2 n
n
∞
[
E = E [sup|f |≤ k],E [sup|f |≤ k]?E [sup|f |≤ k +1].
2 2 n 2 n 2 n
n n n
k=1
mE = lim mE [sup | f |≤ k]. k mE ? mE [sup | f |≤ k ] < .
2 2 n 0 2 2 n 0
k→∞
n n
E =E [sup|f |≤ k ],c = k . E n,|f (x)|≤ c,
0 2 n 0 0 0 n
n
m(E?E ) = m(E?E )+m(E ?E ) <.
0 2 2 0
6. f(x) (?∞,∞) g(x) [a,b] f(g(x))
E = (?∞,∞),E = [a,b]. f(x) E c,E [f < c]
1 2 1 1
∞
S
E [f >c] = (α ,β ), (α ,β ) ( α
1 n n n n n
n=1
∞ ∞
S S
?∞,β +∞). E [f(g) > c] = E [α < g < β ] = (E [g > α ]∩E [g < β ]),
n 2 2 n n 2 n 2 n
n=1 n=1
g E E [g > α ],E [g <β ] E[f(g)>c]
2 2 n 2 n
7. f (x),(n = 1,2, ) E ” ” f(x), {f }a.e.
n n
f.
f (x) E ” ” f(x), δ > 0,
n
E ? E, m(E?E ) < δ f E f(x). E E f
δ δ n δ 0 n
δ, E ? E?E ( E f ), mE ≤ m(E?E ) < δ, δ → 0,
0 δ δ n 0 0
mE = 0. f (x) E a.e. f(x)( ).
0 n
2
8.
f(x) E δ > 0, E ? E
δ
f(x) E m(E?E )< δ, f(x) E a.e.
δ δ
1/n, E ? E, f(x) E
n n
1
m(E?E ) < .
n
n
∞ ∞
S S
1
E = E? E , n, mE = m(E? E )≤ m(E?E ) < . n→∞,
0 n 0 n n
n
n=1 n=1
∞ ∞
S S
mE = 0. E = (E?E )∪E = ( E )∪E = E . a,E[f > a] = E [f >
0 0 0 n 0 n 0
n=1 n=0
∞
S
? ?
a]∪( E [f > a]), f E E [f > a] m (E [f > a])≤ m E = 0,
n n n 0 0
n=1
E [f > a] E[f >a] f f E
0 n
∞
S
E f(x)a.e.
n
n=1
9. {f } E f, f (x) ≤ g(x)a.e. E,n = 1,2, .
n n
f(x)≤g(x) E
f (x)?f(x), {f }?{f }, f (x) E a.e. f(x). E
n n n n 0
i i
∞ ∞
S P
f (x) f(x) E = E[f > g]. mE = 0,mE = 0.m( E )≤ mE = 0.
n n n 0 n n n
i
n=0 n=0
∞ ∞
S S
E? E f (x)≤g(x),f (x) f(x), f(x) = limf (x)≤g(x) E? E
n n n n
i i n
i
n=0 n=0
f(x)≤ g(x) E
10. E f (x)?f(x), f (x)≤ f (x) n = 1,2, ,
n n n+1
f (x) f(x).
n
f (x)?f(x), {f }?{f }, f (x) E a.e. f(x). E
n n n n 0
i i
f (x) f(x) E =E[f < f ], mE = 0,mE = 0.
n n n n+1 0 n
i
∞ ∞
[ X
m( E )≤ mE = 0
n n
n=0 n=0
∞
S
E? E f (x) f(x), f (x) f (x) f(x). (
n n n n
i
n=0
∞
S
). E f (x) f(x),
n n
n=0
f (x)a.e. f(x).
n
11. E f (x)?f(x), f (x) = g (x)a.e. n = 1,2, , g (x)?f(x).
n n n n
∞ ∞
S P
E = E[f = g ] m( E )≤ mE = 0. σ > 0,E[| f ?g |≥ σ]?
n n n n n n
n=1 n=1
∞
S
( E )∪E[|f?f |≥ σ].
n n
n=1
∞
[
mE[|f ?g |≥ σ]≤m( E )+mE[|f?f |≥σ] = mE[|f?f |≥σ].
n n n n
n=1
3?????????
???
?
??
f (x)?f(x), 0≤ limmE[|f?g |≥ σ]≤ limmE[| f?f |]≥ σ = 0 g (x)?f(x).
n n n n
12. mE < +∞, E f (x)? f(x) {f }
n n
{f }, {f } {f }, lim f (x) = f(x),a.e. E.
n n n n
k k k k
j j
j→∞
{f (x)} E
n
f(x). η > 0, {mE[| f ?f |≥ η ]} > 0,
0 n 0 0
{f },
n
k
mE[|f ?f |≥ η ] > > 0. (1)
n 0 0
k
{f } f(x) {f }
n n
k k
j
E a.e. f, mE < +∞, E f ? f(x), (1)
n
k
j
13. mE <∞, f (x) g (x),n = 1,2, ,
n n
f(x) g(x),
(1)f (x)g (x)?f(x)g(x);
n n
(2)f (x) +g (x)? f(x)+g(x);
n n
(3)min{f (x),g (x)}? min{f(x),g(x)};max{f (x),g (x)}? max{f(x),g(x)}.
n n n n
(1) f(x)a.e. mE[|f |=∞] = 0.
∞
T
E[| f |≥ n] = E[| f |=∞], E[| f |≥ n]? E[| f |≥ n+1] E[| f |≥ 1]? E,mE[|
n=0
f |≥ 1]≤ mE <∞,
mE[|f |=∞] = lim mE[|f |≥ n] = 0.
n→∞
lim mE[|g|≥n] = 0.
n→∞
> 0,σ > 0, k,mE[|f |≥ k]< mE[|g|≥k]<
5 5
σ
σ = min ,1 , f ?f,g ?g, N, n>N mE[|g ?g|≥ σ ]<
0 n n n 0
2(k+1)
,mE[|f ?f |≥ σ ] <
n 0
5 5
E[|g |≥ k +1]?E[|g|≥ k]∪E[|g ?g|≥ 1]? E[|g|≥ k]∪E[|g ?g|≥ σ ].
n n n 0
2
mE[| g |≥ k +1]≤ mE[|g|≥ k]+mE[| g ?g|≥σ ]< + = .
n n 0
5 5 5
h i
σ σ
E |g f ?g f |≥ ? E[|g |≥k+1]∪E |f ?f |≥ ? E[|g |≥k+1]∪E[|f ?f |≥ σ ].
n n n n n n n 0
2 2(k +1)
h i
σ 2 3
mE |g f ?g f |≥ ≤ mE[|g |≥ k +1]+mE[|f ?f |≥ σ ]< + = .
n n n n n 0
2 5 5 5
h i
σ σ
E |fg ?fg|≥ ?E[|f |≥ k+1]∪E |g ?g|≥ ?E[| f |≥ k]∪E[|g ?g|≥σ ].
n n n 0
2 2(k +1)
4????
??????
h i
σ 2
mE | fg ?fg|≥ ≤ mE[|f |≥ k]+mE[| g ?g|≥ σ ]< + = .
n n 0
2 5 5 5
h i h i
σ σ
E[|g f ?gf |≥ σ]? E |g f ?g f |≥ ∪E |fg ?fg|≥ ,
n n n n n n
2 2
h i h i
σ σ 3 2
mE[|g f ?gf |≥σ]≤mE | g f ?g f |≥ +mE |fg ?fg|≥ < + =.
n n n n n n
2 2 5 5
> 0,σ > 0, N, n>N mE[|g f ?gf |≥σ] < , g f ? gf.
n n n n
(2)
h i h i
σ σ
E[| (f +g )?(f +g)|≥ σ]? E |f ?f |≥ ∪E |g ?g|≥
n n n n
2 2
h i h i
σ σ
mE[| (f +g )?(f +g)|≥ σ]≤mE |f ?f |≥ +mE |g ?g|≥ ,
n n n n
2 2
h i h i
σ σ
lim mE[| (f +g )?(f +g)|≥ σ]≤ lim mE |f ?f |≥ + lim mE |g ?g|≥ .
n n n n
n→∞ n→∞ 2 n→∞ 2
f +g ?f +g.
n n
(3) f ?f, |f |?|f |.
n n
E[|f ?f |≥σ]? E[||f |?|f ||≥ σ].
n n
lim mE[||f |?|f ||≥ σ]≤ lim mE[|f ?f |≥σ] = 0, |f |?|f |.
n n n
n→∞ n→∞
f ?f, a = 0,af ?af.
n n
σ
E[|af ?af |≥ σ] = E |f ?f |≥ ,
n n
| a|
σ
lim mE[|af ?af |≥ σ] = lim mE |f ?f |≥ = 0.
n n
n→∞ n→∞
|a|
f (x)+g (x)?|f (x)?g (x)|
n n n n
min{f (x),g (x)} = .
n n
2
(2),
f (x)+g (x)?f +g,f (x)?g (x)?f?g.
n n n n
| f (x)?g (x)|?|f(x)?g(x)|,
n n
(2),
f (x)+g (x)?|f (x)?g (x)|?f(x)+g(x)?| f(x)?g(x)|.
n n n n
f (x)+g (x)?|f (x)?g (x)| f(x)+g(x)?| f(x)?g(x)|
n n n n
? .
2 2
5
min{f (x),g (x)}? min{f(x),g(x)}.
n n
f (x)+g (x)+|f (x)?g (x)|
n n n n
max{f (x),g (x)} = ,
n n
2
max{f (x),g (x)}? max{f(x),g(x)}.
n n
6??
1. Lebesgue Darboux
Darboux f (x) E E
D : E , E , , E , max mE → 0
1 2 n i
1≤i≤n
ˉ
S (D, f )→ f (x)dx, S (D, f )→ f (x)dx.
E
E
?
[0, 1]
?
?
?
? 1, x [0,1] ,
?
f (x)=
?
?
?
?
0, x [0,1] .
n n i?1 i n n?1
n, [0,1] D = {E }, E = , , i= 1, 2, , n? 1, E = , 1 .
n
n
i i n n n
1
n
max mE = → 0(n→∞).
i n
1≤i≤n
n n
1
n
S (D, f )= sup mE = 1 = 1.
i
n
n
x∈E
i=1 i i=1
f (x) [0,1]
ˉ
f (x)dx= f (x)dx= 0.
[0,1] [0,1]
Darboux
1
2. Cantor P f (x) = 0, P
0 0 n
3
n(n= 1, 2, ), f (x)
f (x) E P
n 0
n?1
1 2
mE = ,
n n n
3 3
∞ ∞ ∞
n?1
2
f (x)dx= f (x)dx= nmE = n = 3.
n
n
3
[0,1] E
n
n=1 n=1 n=1
f (x) 3.
3. f (x) E e = E[| f |≥ n],
n
lim n me = 0.
n
n
f (x) E E a.e. mE[| f |=∞]= 0.
∞
e ? e , me ≤ mE <∞ e = E[| f |=∞],
n n+1 1 n
n=1
lim me = mE[| f |=∞]= 0.
n
n
1?????????
| f (x)| > 0, δ > 0, e? E me < δ
| f (x)| dx < .
e
δ > 0, N, n > N me < δ,
n
n me ≤ | f (x)| dx < .
n
e
n
lim n me = 0.
n
n
4. mE <∞, f (x) E E = E[n? 1≤ f < n], f (x) E
n
∞
| n| mE <∞.
n
?∞
f (x) E | f (x)| E
n≥ 1 E n? 1≤| f (x)|= f (x) < n.
n
n≤ 0 E | n|≤| f |≤| n? 1|= 1? n,
n
∞ ?∞ ∞ ?∞
∞ > | f (x)| dx= | f | dx+ | f | dx≥ (n? 1)mE + | n| mE
n n
E E E
n n
n=1 n=0 n=1 n=0
∞ ?∞ ∞ ∞ ∞
= | n| mE + | n| mE ? mE = | n| mE ? mE ,
n n n n n
?∞
n=1 n=0 n=1 n=1
E E
n
∞ ∞
mE = m( E )≤ mE <∞,
n n
n=1 n=1
∞
| n| mE <∞.
n
?∞
∞
| n| mE ,
n
?∞
∞ ?∞ ∞ ?∞
| f (x) | dx= | f | dx+ | f | dx≤ nmE + | n? 1| mE
n n
E E E
n n
n=1 n=0 n=1 n=0
∞ ?∞ ?∞ ∞
= | n| mE + | n| mE + mE ≤ | n| mE + mE <∞.
n n n n
?∞
n=1 n=0 n=0
+ ? + ?
| f (x)| f (x) f (x) f (x)= f (x)? f (x)
5. f (x) [a, b] R ( ), f (x) [a, b] L
| f (x)| [a, b] R ( ) ,
b
f (x)dx= (R) f (x)dx.
[a,b] a
f (x) [a, b] R ( ), 0 < < b ? a, f (x)
[a+ , b] R | f (x)| [a+ , b] R §2 4
b
(R) | f (x)| dx= | f (x)| dx.
a+ [a+, b]
2??
→ 0,
b
(R) | f (x)| dx= | f (x) | dx.
a [a,b]
f (x) [a, b] L | f (x)| [a, b] §5
7,
b b b
+ ? + ?
f (x)dx= f (x)dx? f (x)dx= (R) f (x)dx? (R) f (x)dx= (R) f (x)dx.
[a,b] [a,b] [a,b] a a a
6. { f } E lim f (x)dx= 0, f (x)? 0.
n n n
E
n→∞
σ > 0, f
n
σmE[| f |≥ σ]≤ f (x)dx≤ f dx.
n n n
E[| f |≥σ] E
n
1
mE[| f |≥ σ]≤ f (x)dx,
n n
σ
E
1
lim mE[| f |≥ σ]= lim f (x)dx= 0.
n n
n→∞
σ
E
f (x)? 0.
n
7. mE <∞,{ f } a.e.
n
| f (x)|
n
lim dx= 0
n→∞
1+| f (x)|
n
E
f (x)? 0.
n
f ? 0
n
| f |
n
E ≥ σ ? E[| f |≥ σ],
n
1+| f |
n
| f |
n
lim mE ≥ σ ≤ lim mE[| f |≥ σ]= 0.
n
1+| f |
n
| f | | f |
n n
? 0, 0≤ < 1(n= 1, 2, ), mE <∞.
1+| f | 1+| f |
n n
| f (x)|
n
lim dx= 0dx= 0.
n→∞
1+| f (x)|
n
E E
| f (x)| | f (x)|
n n x
lim dx = 0, 6 dx ? 0. y = ,
1+| f (x)| 1+| f (x)| 1+x
E n E n
n→∞
x >?1
| f | σ
n
E ≥ = E[| f |≥ σ],
n
1+| f | 1+σ
n
| f |
n σ
lim mE[| f |≥ σ]= lim mE ≥ = 0, f ? 0.
n n
1+| f | 1+σ
n
n→∞ n→∞
1
sin
x
8. f (x)= , 0 < x ≤ 1, α f (x) [0,1] L
α
x
3
α≥ 1
1
1 1 ∞
π
1 1 1 1 sin y
| f (x)| dx≥ | sin | dx≥ | sin | dx= | | dy=∞,
x x x x y
0 0 0 π
α≥ 1 f (x)L
1
sin
1 x 1
α < 1 [0,1] ≤ , f (x)L
α α α
x x |x |
9. [0,1] n E , E , , E , [0,1] n
1 2 n
q q/n.
? (x) E [0,1] q [0,1]
i i
n
? (x)≥ q. mE = ? (x)dx,
i i i
[0,1]
i=1
n n
mE = ? (x)dx≥ qdx= q.
i i
[0,1] [0,1]
i=1 i=1
q
E , mE ≥ .
i i
n
10. mE, 0, f (x) E ?(x),
f (x)?(x)dx= 0,
E
f (x)= 0 a.e. E.
δ > 0, ?(x) E[ f > δ]
δmE[ f ≥ δ]≤ f (x)dx= f (x)?(x)dx= 0,
E[ f≥δ] E
mE[ f ≥ δ]= 0. mE[ f ≤?δ]= 0, mE[| f|≥ δ]= 0.
∞
1
E[ f , 0]= E | f|≥ .
n
n=1
∞
1
mE[ f , 0]≤ mE | f|≥ = 0, f (x)= 0 a.e. E.
n
n=1
11.
dt
lim = 1.
n
1
t
n
n
(0,∞) 1+ t
n
t ∈ (0, 1)
1 1 1
≤ ≤ (n > 2);
√
n 1 1
t
n t
n t
1+ t
n
t ∈ [1,∞) n > 2
1 1 2n 4
= < < .
n
1 1
2 2
t n?1
2
t (n? 1) t
n n
1+ t 1+ t+ t + t
2n
n
4′′′′
?
1
?
?
?
? , t ∈ (0, 1),
√
?
?
?
t
F(t)=
?
?
?
? 4
?
?
, t ∈ [1,∞),
?
2
t
1 ∞
dt 4dt
F(x)dx= √ + = 6,
2
t
t
(0,∞) 0 1
F(x) (0,∞)
dt 1 dt
lim = lim dt= = 1.
n 1 n 1
t
n t n t
e
n n
(0,∞) (0,∞) (0,∞)
1+ t 1+ t
n n
1 2 3
12. = (1? x)+ (x ? x )+ , 0 < x < 1,
1+x
1 1 1
ln 2= 1? + ? + .
2 3 4
n n+1
[0,1] x ? x ≥ 0, §5 3
∞ ∞
1 1
1 1 1 1 1 1
2n 2n+1
dx= (x ? x )dx= ? = 1? + ? + ,
1+ x 2n+ 1 2n+ 2 2 3 4
0 0
n=0 n=0
1
1 1 1 1
dx= ln 2, ln 2= 1? + ? + .
1+x 2 3 4
0
13. f (x, t) |t? t | < δ x [a, b] K,
0
?
f (x, t) ≤ K, a≤ x≤ b, |t? t | < δ,
0
?t
b b
d
f (x, t)dt= f (x, t)dx.
t
dt
a a
h , lim h = 0 h , 0.
n n n
n→∞
b b
d 1
f (x, t)dt= lim [ f (x, t+ h )? f (x, t)]dx,
n
dt n→∞ h
a n a
f (x, t+ h )? f (x, t) f (x, t+θh ) h
n n n
t
= =| f (x, t+θh )|≤ K,
n
t
h h
n n
0 < θ < 1, a≤ x≤ b, t ?δ < t+θh < t +δ.
0 n 0
b b b
d f (x, t+ h )? f (x, t)
n
f (x, t)dt= lim dx= f (x, t)dx.
t
n→∞
dt h
n
a a a
14.
∞
1 p
x 1 1
ln dx= (p >?1).
2
1? x x (p+ n)
0
n=1
5?′′′′′?????′?′?′′???
∞ ∞
p
x 1 1 1
n p n+p
ln = ( x )x ln = x ln ,
1? x x x x
n=0 n=0
1
n+p
x∈ (0, 1) x ln ≥ 0,
x
∞ ∞ ∞
1 1
p
x 1 1 1
n+p
ln dx=? x ln xdx= = .
2 2
1? x x (n+ p+ 1) (n+ p)
0 0
n=0 n=0 n=1
15. { f } E lim f (x)= f (x)a.e. E,
n n
n→∞
| f (x)|dx < K, K ,
n
E
f (x)
| f (x)|dx= lim | f (x)|dx≤ lim | f (x)|dx≤ K,
n n
n→∞
E E n→∞ E
| f (x)| f (x)
16. f (x) [a? , b+ ]
b
lim | f (x+ t)? f (x)|dx= 0.
t→0
a
§4 1, σ > 0, [a? , b+ ] ?(x),
b+
σ
| f (x)??(x)|dx < .
3
a?
?(x) [a?, b+ ] δ > 0( δ < ), x , x ∈ [a?, b+ ],
|x ? x | < δ,
σ
|?(x )??(x )| < .
3(b? a)
0 < t < δ x ∈ [a, b],
σ
|?(x+ t)??(t)| < ,
3(b? a)
b b b b
| f (x+ t)? f (x)|dx≤ | f (x)??(x)|dx+ | f (x+ t)??(x+ t)|dx+ |?(x+ t)??(x)|dx
a a a a
σ σ σ
< + + (b? a)= σ.
3 3 3(b? a)
b
lim | f (x+ t)? f (x)|dx= 0.
t→0
a
6′′′???′′???′′′′??
17. f (x), f (x)(n= 1, 2, ) E lim f (x)= f (x)a.e. E,
n n
n→∞
lim | f (x)|dx= | f (x)|dx,
n
n→∞
E E
e? E,
lim | f (x)|dx= | f (x)|dx.
n
n→∞
e e
| f (x)|dx= lim | f (x)|dx≤ lim | f (x)|dx.
n n
e e e
n→∞ n→∞
| f (x)|dx≥ lim | f (x)|dx.
n
n→∞
e e
| f (x)|dx < lim | f (x)|dx,
n
n→∞
e
{ f (x)},
n
i
lim | f (x)|dx= lim | f (x)|dx > | f (x)|dx,
n n
i
n→∞
i→∞
e e e
lim | f (x)|dx= lim | f (x)|dx? lim | f (x)|dx < | f (x)|dx? | f (x)|dx= | f (x)|dx,
n n n
i i i
i→∞ i→∞ i→∞
E?e E e E e E?e
lim | f (x)|dx≥ | f (x)|dx≥ lim | f (x)|dx,
n n
n→∞
n→∞ e e e
lim | f (x)|dx= | f (x)|dx.
n
n→∞
e e
18. f (x) (0,∞)
lim f (x)= 0.
x→∞
f (x) (0,∞) | f (x)| (0,∞) lim f (x), 0 , > 0
0
x→∞
x ∈ (0,∞), lim x = ∞, | f (x )| ≥ . f (x) (0,∞) δ > 0,
n n n 0
n→∞
0
x , x ∈ (0,∞), |x ? x | < δ | f (x )? f (x )| < . x ∈ (x ?δ, x +δ)
n n
2
0
| f (x)? f (x )| < .
n
2
0 0
| f (x)|≥| f (x )|? ≥ .
n
2 2
x →∞, {x }, x < x , |x ?x | > 2δ, i= 1, 2, . E = (x ?δ, x +δ),
n n n n n n n n n
i i i+1 i+1 i i i i
{E }
n
i
∞ ∞ ∞
0
| f (x)|dx≥ | f (x)|dx= | f (x)|dx≥ mE = δ =∞,
n 0
∞ i
2
(0,∞) E E
n n
i i=1 i i=1 i=1
i=1
7
| f (x)| (0,∞) lim f (x)= 0.
x→∞
p q p q
19. f (x) R g(y) R f (x) g(y) R × R
p p p q
f (x) R R R × R
p q p q
g(y) R × R f (x)g(y) R × R
p q
f (x), g(y) §6 4, f (x)g(y) R × R
f (x)g(x)dxdy= dx f (x)g(y)dy= f (x)dx g(y)dy <∞,
p q p q p q
R ×R R R R R
+ ? + ? + ? +
f (x), g(x) f (x)= f (x)? f (x), g(y)= g (y)?g (y), f (x), f (x), g (y),
? + + ? ? ? + + ?
g (y) f (x)g(y)= f (x)g (y)+ f (x)g (y)? f (x)g (y)? f (x)g (y)
f (x)g(y)
20. D :?1≥ x≥ 1,?1≥ y≥ 1
?
xy
2 2
?
?
, x + y , 0,
?
? 2 2 2
?
(x + y )
f (x)=
?
?
?
?
?
0, x= y= 0,
f (x, y) f (x, y) D
1
xy
x, y f (x, y) dx
2 2 2
?1 (x +y )
1
xy
dy
2 2 2
(x +y )
?1
1 1 1 1
xy xy
dy dx= dx dy= 0.
2 2 2 2 2 2
(x + y ) (x + y )
?1 ?1 ?1 ?1
f (x, y) D f (x, y) D : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
1
1 1
xy
dy dx
2 2 2
(x +y )
0 0
1
xy 1 y
dx= ?
2 2 2 2
(x + y ) 2y 2(y + 1)
0
[0,1] f (x, y) D
21. f (x), g(x) E f (x)g(x) E E = E[g≥ y].
y
F(y)= f (x)dx
E
y
y > 0
+∞
F(y)dy= f (x)g(x)dx.
0 E
f (x), g(x) E y > 0,E F(y)= f (x)dx
y
E
y
( +∞) F(y)≥ 0.
? ?
+∞ +∞ ∞ g(x)
? ?
? ?
? ?
? ?
F(y)dy= f (x)dx dy= χ (x) f (x)dx dy= f (x) dy dx= f (x)g(x)dx.
? ? E
y
0 0 E 0 E E 0 E
y
8??
1. (a,b)
f(x),g(x) (a,b) E (a,b) E f(x) =g(x).
M f(x) N g(x)
x ∈ M, f(x +0) = f(x ) = f(x ? 0). E (a,b) x ∈E,
0 0 0 0 n
x ≤x lim x =x ,
n 0 n 0
n→∞
f(x ?0) = lim f(x ) = lim g(x ) =g(x ?0).
0 n n 0
n→∞ n→∞
f(x +0) =g(x +0), x g(x) x ∈E, M ?N.
0 0 0 0
N ?M, M =N, f(x) g(x)
2. {f } [a,b] f (x) → f(x)(n→∞), f(x)
n n
b
(f )n
a
[a,b] T :a =x 0 1 m
m b
|f (x )?f (x )|≤ (f )n i n i?1 n
a
i=1
m m
|f (x )?f (x )| = lim |f (x )?f (x )|≤K.
n i n i?1 n i n i?1
n→∞
i=1 i=1
b
f(x) [a,b] (f)≤K.
a
α 1
3. f(x) =x sin (0≤x≤ 1;α,β > 0)
β
x
α≤β, [0,1]
1
?
π β
x = 0,x = ((n?1)?k)π + ,k = 1,2,··· ,n?1,x = 1.
0 k n
2
n n
1 1
α α
|f(x )?f(x )| = x sin ?x sin
k k?1
k k?1
β β
x x
k k?1
k=1 k=1
α α
n?1
β β
1 1
≥ +
π π
((n?1)?k)π + (n?k)π +
2 2
k=1
n?1
1 1
≥ +
π π
((n?1)?k)π + (n?k)π +
2 2
k=1
n?1
1 1
> +
(n?k)π (n?k +1)π
k=1
n?1 n
2 2 1
= .
>
(n?k +1)π π k
k=2
k=1
1?′′′′′′′′′′′′
?′′
n
1
=∞,
k
k=2
1 ∞
(f) = sup |f(x )?f(x )| =∞,
k k?1
0 k=1
f(x) [0,1]
α>β > 0.
1 1
α?1 α?β?1
f (x) =αx sin ?βx cos ,
β β
x x
α?1 α?β?1
|f (x)|≤αx +βx .
α>β > 0 α?1> 1,α?β?1>?1. |f (x)| [0,1] R L
f (x) [0,1]
x
x x x
1 1
α α
(L) f (t)dt = lim(L) f (t)dt = lim(R) f (t)dt = lim t sin =x sin ?0 =f(x)?f(0).
β β
δ→0 δ→0 δ→0
t x
0 δ δ
δ
x
f(x) =f(0)+ f (t)dt.
0
f(x) [0,1]
4. f(x) [a,b] f (x)≥ 0a.e. [a,b], f(x)
x ,x ∈ [a,b],x >x . f(x) [a,b]
1 2 2 1
x
1
f(x ) =f(a)+ f (t)dt;
1
a
x
2
f(x ) =f(a)+ f (t)dt.
2
a
x
2
f(x )?f(x ) = f (t)dt.
2 1
x
1
x
2
f (x)≥ 0a.e. [a,b], f (t)dt≥ 0, f(x )≥f(x ), f(x)
2 1
x
1
5. f(x) [a,b] M > 0, > 0
b
(f)≤M,
a+
f(x) [a,b]
?x∈ (a,b),
b
|f(x)?f(b)|≤ (f)≤M,
x
|f(x)|≤M +|f(b)|, [a,b] T,
T :a =x 0 1 2 n
2′′′′′′′′′
T
n n
v = |f(x )?f(x )| =|f(x )?f(a)|+ |f(x )?f(x )|
i i?1 1 i i?1
i=1 i=2
b
≤|f(x )|+|f(a)|+ (f)≤ 2M +|f(b)|+|f(a)|,
1
x
1
b
(f)≤ 2M +|f(b)|+|f(a)|<∞,
a
f(x) [a,b]
∞
6. {f } [a,b] f(x) = f (x) [a,b]
n n
n=1
f(x) [a,b]
n, f (x) [a,b]
n
x
f (x) =f (a)+ f (t)dt.
n n
n
a
f (x) f (x)≥ 0,a.e. [a,b]. f(x) [a,b]
n
n
f (x) [a,b] L
∞ ∞ ∞
x
f(x) = f (x) = f (a)+ f (t)dt.
n n
n
a
n=1 n=1 n=1
∞ ∞
x x
f (t)dt = f (t)dt.
n n
a a
n=1 n=1
∞
f (x)
n
n=1
∞ ∞
b
f (t)dt≤ [f (b)?f (a)] <∞.
n n
n
a
n=1 n=1
∞
f (x) [a,b] L
n
n=1
∞ ∞
x
f(x) = f (a)+ f (t)dt
n
n
a
n=1 n=1
[a,b]
7. f(x) [a,b]
(1)f(x) [a,b] Lipschitz
(2)f(x) [a,b]
(2) (1).
x
f(x) = g(t)dt,
a
3′′′?′′′′′′′′
′′?′′′′′′′′′
g(x) [a,b] |g(x)|≤K,x∈ [a,b]. [a,b] x,x , x >x ,
x
|f(x )?f(x )| = g(t)dt ≤K|x ?x |,
x
f(x) [a,b] Lipschitz
(1) (2). f(x) [a,b] Lipschitz f(x) [a,b]
x
f(x) =f(a)+ f (t)dt.
a
x,y∈ [a,b],
x
f (t)dt =|f(x)?f(y)|≤K|x?y|,
y
K Lipschitz |f (x)|≤K,a.e. [a,b].
?
?
f (x), |f (x)|≤K,
g(x) =
?
K, |f (x)| >K,
g(x) [a,b]
x
f(x) =f(0)+ g (t)dt.
0
8. ” ” S
” ” f(x),α(x) [a,b] [a,b]
T :a =x 0 1 n
M m f(x) [x ,x ] i = 1,2,··· ,n,
i i i?1 i
n
S(T,f,α) = M (α(x )?α(x )),
i i i?1
i=1
n
s(T,f,α) = m (α(x )?α(x )).
i i i?1
i=1
ˉ
b
f(x)dα(x) = infS(T,f,α),
T
a
b
f(x)dα(x) = sups(T,f,α),
T
a
ˉ
b b
f(x)dα(x) = f(x)dα(x), f(x) α(x)S
a
a
” ” ” ”
?
1 1
?
?0, x∈ ?1,? ∪ ,1 ,
?
2 2
f(x) =
? 1 1
?
?
1, x∈ ? , ,
2 2
4
?
?
1 1
?
?0, x∈ ?1,? ∪ ,1 ,
?
2 2
α(x) =
? 1 1
?
?
1, x∈ ? , .
2 2
1 1
[?1,1] T :?1 = x <··· < x 0 i?1 i j?1 j n
2 2
n
σ = f(ξ )[α(x )?α(x )] =f(ξ )?f(ξ )
i i i?1 i j
i=1
?
1 1
?
?1, ξ >? ,ξ > ,
i j
?
?
2 2
?
?
?
1 1 1 1
=
0, ξ >? ,ξ < ξ ,
i j i j
?
2 2 2 2
?
?
?
?
1 1
?
?
?1, ξ i j
2 2
σ f(x) [?1,1] α(x) S
” ” f(x) S T :?1 = x < x <··· < x = 1,
0 1 n
1 1
? , (x ,x ) α(x ) = α(x ) = ··· = α(x ) = 0. S(T,f,α) =
i?1 i 0 1 n
2 2
1 1 1 1
0,s(T,f,α) = 0; ? , ? ∈ [x ,x ], ∈ [x ,x ],
i?1 i j?1 j
2 2 2 2
n
S(T,f,α) = M (α(x )?α(x )) = 1?1 = 0,
k k k?1
k=1
n
s(T,f,α) = m (α(x )?α(x )) = 0,
k k k?1
k=1
1
1
ˉ
f(x)dα(x) = f(x)dα(x), f(x) S
?1
?1
8. α(x) (?∞,∞) ( )
L?S
1
E ?R ,
∞ ∞
? ?
m E = inf m (a ,b ), (a ,b )?E .
i i i i
α α
i=1 i=1
α α(b )≥α(b ?0),α(a )≤α(a + 0),
i i i i
?
|I | =α(b )?α(a )≥α(b ?0)?α(a +0) =m (a ,b ),
i i i i i i i
α
∞ ∞ ∞ ∞
?
I = (a ,b )?E, m (a ,b )≤ |I|,
i i i i i i
α
i=1 i=1 i=1 i=1
∞ ∞
? ?
inf m (a ,b ) ≤ inf |I| =m E,
i i i
α α
i=1 i=1
5′′′′′??′′′
∞
(a ,b )?E,
i i
i=1
∞ ∞
? ? ?
m (a ,b )≥m (a ,b ) ≥m E,
i i i i
α α α
i=1 i=1
∞ ∞
? ?
inf m (a ,b ), (a ,b )?E ≥m E.
i i i i
α α
i=1 i=1
α(x) α (x).
? ?
x, α(x?0) =α (x?0),α(x+0) =α (x+0), m (a ,b ) =m (a ,b ).
i i i i
α α
1
E ?R ,
∞ ∞
? ?
m E = inf m (a ,b ), (a ,b )?E
i i i i
α α
i=1 i=1
∞ ∞
= inf (α(b ?0),α(a +0)), (a ,b )?E
i i i i
i=1 i=1
∞ ∞
= inf (α (b ?0),α (a +0)), (a ,b )?E
i i i i
i=1 i=1
∞ ∞
?
= inf m (a ,b ), (a ,b )?E
i i i i
α
i=1 i=1
?
=m E.
α
L?S L?S α L?S L?S
L?S α(x)
L?S
6 |
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