《实变函数》第一章 集合论与点集论习题选解
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E Ec, E c = {x : ε > 0 f (x + ε) − f (x − ε) ≤ 0} f (x + ε) − f (x − ε) ≥ 0, E c = {x : ε > 0 f (x + ε) − f (x − ε) = 0} Ex 14: F ⊂ Rn F ⊂ Rn F E F E, E ∩ F = ∅, E ∩ F = ∅. F F ⊂ Rn Bolzano-Weierstrass F ⊂ Rn E E = ∅, F Ex 15: F ⊂ Rn r > 0, E = {t ∈ Rn : x ∈ F, d(t, x) = r}
j →∞ ∞
=
k=1 ∞
x : lim fj (x) ≥
j →∞
1 k 1 k 1 k
Fra Baidu bibliotek
=
k=1 ∞
x : lim sup fj (x) ≥
n→∞ j ≥n ∞ ∞
Ex 2:
9@ c EGF
{fn (x)}
!
= [a, b]
n→∞
k=1 n=1 j =n
"#$%&'
x : fj (x) ≥
# v
{(x, y ) : x2 + y 2 = r},
r ∈[0,1)
{(x, y ) : x2 + y 2 = r},
1
! xy % 6 z8 ({|2B ~ 6
Ex 7: f (x) x1 , x 2 , · · · , x n , k, Ex 9: E R3
E ⊂ [a, b]
78
lim fn (x) = χ[a,b]\E (x), x ∈ [a, b].
56DC
n→∞
1 En = {x ∈ [a, b] : fn (x) ≥ 2 },
(A2B
n→∞
lim En .
x ∈ limn→∞ En , ∃N , n ≥ N limn→∞ En ; x ∈ lim En , ∀n, ∃k ≥ n,
Ex 18:
f ∈ C (R1 ),{Fk }
R1
Xn c, (0, 0, · · · , 0, x∗ , 0 , · · · ) ∈ / Pn (Dn ), n
∞
Dn < c, Pn (Dn ) ≤ Dn < c, ∀n, ∃x∗ n, ∗ ∗ ∗ (x1 , x2 , · · · , xn , · · ·) ∈ / Dn , (x1 , x2 , · · · , x∗ / n , · · ·) ∈ Dn0 = c, An0 = c.
k=1 1 {x : f (x) > k } 1 {x : f (x) < − k }
E = {x ∈ [0, 1] : f (x) = 0} =
{x : f (x) >
1 } k
∞
1 {x : f (x) < − } k
E
E
x0 ∈ E ,
E ⊂
S (x0 , r),
Q+
S (x0 , r)
r ∈Q+
QUV ' I WX (de&fghiprq
1, x ∈ [a, b] \ E 0, x∈E
x ∈ En , fn (x) ≥ 1/2, [a, b] \ E ⊂ fk (x) ≥ 1/2, E ∩ lim En = ∅.
n→∞
RGS Y`ab
56 X tt uu v ') wxy w s ( 56) 5d e 5d f g hiyjkuwwlmv
n=1
f:
An → [0, 1]∞
[0, 1]∞
f (An ) = Dn ⊂ [0, 1]∞ , An ∼ Dn . Xn = {(0, 0, · · · , 0, xn , 0, · · ·) : xn ∈ [0, 1]}
Pn : Dn → Xn , Pn ((x1 , x2 , · · · , xn , · · ·)) = (0, 0, · · · , 0, xn , 0, · · ·)
r ∈Q+
C (x1 , r ) ∩ C (x2 , r ) Ex 12: E=
∞
An ,
E = c,
n0 ,
An0 = c.
n=1
[0, 1]∞ = {(x1 , x2 , · · · , xn , · · ·) : xn ∈ [0, 1]},
∞
'» T ¼ d½¾ yº § ) y¶· ¸ y¹º ) v·
n→∞
F
lim En = [a, b] \ E .
HGI T P
n→∞
lim fn (x) = χ[a,b]\E (x) =
Ex 4: f : X → Y, A ⊂ X, B ⊂ Y , (i)f −1 (Y \ B ) = f −1 (Y ) \ f −1 (B ); (ii)f (X \ A) = f (X ) \ f (A). (i)
UÃV
Dn = [0, 1]∞ ,
n0 ,
n=1
Ex 13:
f (x)
R1
E = {x :
ε > 0 f (x + ε) − f (x − ε) > 0}
R1
2
~ 6 Î E c É Ê Y`Ï ÐÑÒÓÔ WX c ÉÊ Ò × Ø l ÛÝÜÝTÞÝÉÏÝÊ WÝX UV Õ Ö Ï Ê Ù Ú ¥ SÝ c e « ß ' Æ d ' W X Å HI à ã 2'({| 7 t # v 8 áâ x2ã ' 2 8 # yâx( 9 {| 8á 2 v ä ' cGåGe ' w ±GæGGç ' WGX cGåGæG ' C ~ 6 · E Ww X æGGç c Å « ¥ F æÃí ¦ é l  cÃðà G F æGc å ' TG è hGv ±Ã G ê ë ) ' ñ ' SÃìÃÃw f Uà à î ï à ¼ à T à ò ó × ± c ± ö ' õ c ÙÚòô Å « Xgìw Ç æí c¦ å T yG ÷ ø È ' ù WwGX ú lGG v TGÉGÊ GæGí ¦ UGV C e « IV Å H v 2' ({|12 p! 2 v ~ 6 · F TÉÊ c ± ¸û ¦ UV ± ÿ C E ɦ Ê Ud ' ' ' V TC ¨ü uy ý T Å þ ¡ Ü y ¡ ¶ ¢ E I RS æ ± C ' » ü y ý Y ¥ ` ¦¡§ T í £¦ C ' C e E ç¦ Â¡ E cåæ¡ ' ' É Ê ÷ ø G y ¡ ¶ ¡ ¤ ©¡¡¡ y ½ s ' é ¡ ¨ ÷ ø · T c 6 ½¾Â F e ' ST Á e ' S d ¡ ¦¡ d v # ¡¡2&')({| u¡
(ii) Ex 5: {(x, y ) : x2 + y 2 < 1} {(x, y ) : x2 + y 2 < 1} {(x, y ) : x2 + y 2 ≤ 1} [0, 1) [0, 1]
r ∈[0,1]
f (x) = x2 , X = [−1, 1], Y = [0, 1], A = [0, 1]. {(x, y ) : x2 + y 2 ≤ 1}
n=1
An ∼ [0, 1]∞ .
An
E
∞
ww ¿À ' · T S Á¿À C õ d WÃX ÃÄ T WX à « Å Æ ÇÈ ' WXÉÊ UV Å« ! "#ËÌ"Í$%')({|12 t vw # 8 u#2v
t0 ∈ E , E tn , tn → t 0 , tn ∈ E xn ∈ F d(tn , xn ) = r. xn N xn xk 0 , d(tn , xk0 ) = r, (n ≥ N ) d(t0 , xk0 ) = r, t0 ∈ E . xn d(xn , t0 ) ≤ d(xn , tn )+ d(tn , t0 ) = r + d(tn , t0 ), tn → t 0 , xn Bolzano-Weierstrass xn xn → x 0 , F x0 ∈ F , d(t0 , x0 ) = r. ( ) d(x0 , t0 ) ≤ d(x0 , xn ) + d(xn , tn ) + d(tn , t0 ) d(x0 , t0 ) ≥ d(xn , tn ) − d(x0 , xn ) − d(tn , t0 ). S (x, r) F ⊂ Rn r > 0, S (x, r) x r
x∈F
x0 ∈ E c , ε>0 f (x0 + ε) − f (x0 − ε) = 0, ( f (x0 −ε, x0 +ε) ) ∀y ∈ (x0 −ε, x0 +ε), δ = min{y −(x0 −ε), (x0 +ε)−y )}, f (y + δ ) − f (y − δ ) = 0, (x0 − ε, x0 + ε) ⊂ E c , x0 Ec Ec E
Ex 1.
{fj (x)}
¢¡¢£ ¤¢¥¢¦¢§¢¨¢¤¢¦© ! "#$%&')(012
Rn {x : fj (x) ≥ 1 }, (j, k = 1, 2, · · ·) k x : lim fj (x) > 0 .
j →∞
34 12 56
x : lim fj (x) > 0
[0, 1]
"#no$%' 7p!q % %2v }
∞ k=1
M,
rs t
[0, 1]
uvw8
|f (x1 ) + f (x2 ) + · · · + f (xn )| ≤ M,
E = {x ∈ [0, 1] : f (x) = 0}
C EE l g c v 2 v u #12' 7 u # vw 1# 8 %'({| } % ~ 6 G GGG c GG ' G T d¥¦§ ' WX¨© ~ ld ª d S¥ d ')« d '¢¡ £X¤ ' S « ¬GwG c °± ®¯ v 9 ({| p! rs ~ 6 ² c³´µ T
x0
r S (x0 , r) ∩ E ⊂
C (x1 , r ),
C (x1 , r )
x1
r,S (x0 , r) ∩ E r
r ∈Q+
x1 ∈ S (x0 , r) ∩ E ,
C (x1 , r ) ∩ E ⊂ E
C (x1 , r ) ∩ C (x2 , r ), x2 ∈ C (x1 , r ) ∩ E ,
F E E ⊂ F ⊂ F, E ∩ F = E = ∅. E, E ∩ F = ∅, F x1 ∈ F , O(x1 , 1) F c x2 ∈ O(x1 , 1) ∩ F , O(x1 , max{2, d(x1 , x2 )}), F x1 , x2 , · · ·, E = {x1 , x2 , · · ·}, E = ∅, E ∩ F = ∅, F x0 ∈ F , F xn xn → x 0 , E = {xn : n = 1, 2, · · ·}, E ∩F =∅ x0 ∈ F , F
j →∞ ∞
=
k=1 ∞
x : lim fj (x) ≥
j →∞
1 k 1 k 1 k
Fra Baidu bibliotek
=
k=1 ∞
x : lim sup fj (x) ≥
n→∞ j ≥n ∞ ∞
Ex 2:
9@ c EGF
{fn (x)}
!
= [a, b]
n→∞
k=1 n=1 j =n
"#$%&'
x : fj (x) ≥
# v
{(x, y ) : x2 + y 2 = r},
r ∈[0,1)
{(x, y ) : x2 + y 2 = r},
1
! xy % 6 z8 ({|2B ~ 6
Ex 7: f (x) x1 , x 2 , · · · , x n , k, Ex 9: E R3
E ⊂ [a, b]
78
lim fn (x) = χ[a,b]\E (x), x ∈ [a, b].
56DC
n→∞
1 En = {x ∈ [a, b] : fn (x) ≥ 2 },
(A2B
n→∞
lim En .
x ∈ limn→∞ En , ∃N , n ≥ N limn→∞ En ; x ∈ lim En , ∀n, ∃k ≥ n,
Ex 18:
f ∈ C (R1 ),{Fk }
R1
Xn c, (0, 0, · · · , 0, x∗ , 0 , · · · ) ∈ / Pn (Dn ), n
∞
Dn < c, Pn (Dn ) ≤ Dn < c, ∀n, ∃x∗ n, ∗ ∗ ∗ (x1 , x2 , · · · , xn , · · ·) ∈ / Dn , (x1 , x2 , · · · , x∗ / n , · · ·) ∈ Dn0 = c, An0 = c.
k=1 1 {x : f (x) > k } 1 {x : f (x) < − k }
E = {x ∈ [0, 1] : f (x) = 0} =
{x : f (x) >
1 } k
∞
1 {x : f (x) < − } k
E
E
x0 ∈ E ,
E ⊂
S (x0 , r),
Q+
S (x0 , r)
r ∈Q+
QUV ' I WX (de&fghiprq
1, x ∈ [a, b] \ E 0, x∈E
x ∈ En , fn (x) ≥ 1/2, [a, b] \ E ⊂ fk (x) ≥ 1/2, E ∩ lim En = ∅.
n→∞
RGS Y`ab
56 X tt uu v ') wxy w s ( 56) 5d e 5d f g hiyjkuwwlmv
n=1
f:
An → [0, 1]∞
[0, 1]∞
f (An ) = Dn ⊂ [0, 1]∞ , An ∼ Dn . Xn = {(0, 0, · · · , 0, xn , 0, · · ·) : xn ∈ [0, 1]}
Pn : Dn → Xn , Pn ((x1 , x2 , · · · , xn , · · ·)) = (0, 0, · · · , 0, xn , 0, · · ·)
r ∈Q+
C (x1 , r ) ∩ C (x2 , r ) Ex 12: E=
∞
An ,
E = c,
n0 ,
An0 = c.
n=1
[0, 1]∞ = {(x1 , x2 , · · · , xn , · · ·) : xn ∈ [0, 1]},
∞
'» T ¼ d½¾ yº § ) y¶· ¸ y¹º ) v·
n→∞
F
lim En = [a, b] \ E .
HGI T P
n→∞
lim fn (x) = χ[a,b]\E (x) =
Ex 4: f : X → Y, A ⊂ X, B ⊂ Y , (i)f −1 (Y \ B ) = f −1 (Y ) \ f −1 (B ); (ii)f (X \ A) = f (X ) \ f (A). (i)
UÃV
Dn = [0, 1]∞ ,
n0 ,
n=1
Ex 13:
f (x)
R1
E = {x :
ε > 0 f (x + ε) − f (x − ε) > 0}
R1
2
~ 6 Î E c É Ê Y`Ï ÐÑÒÓÔ WX c ÉÊ Ò × Ø l ÛÝÜÝTÞÝÉÏÝÊ WÝX UV Õ Ö Ï Ê Ù Ú ¥ SÝ c e « ß ' Æ d ' W X Å HI à ã 2'({| 7 t # v 8 áâ x2ã ' 2 8 # yâx( 9 {| 8á 2 v ä ' cGåGe ' w ±GæGGç ' WGX cGåGæG ' C ~ 6 · E Ww X æGGç c Å « ¥ F æÃí ¦ é l  cÃðà G F æGc å ' TG è hGv ±Ã G ê ë ) ' ñ ' SÃìÃÃw f Uà à î ï à ¼ à T à ò ó × ± c ± ö ' õ c ÙÚòô Å « Xgìw Ç æí c¦ å T yG ÷ ø È ' ù WwGX ú lGG v TGÉGÊ GæGí ¦ UGV C e « IV Å H v 2' ({|12 p! 2 v ~ 6 · F TÉÊ c ± ¸û ¦ UV ± ÿ C E ɦ Ê Ud ' ' ' V TC ¨ü uy ý T Å þ ¡ Ü y ¡ ¶ ¢ E I RS æ ± C ' » ü y ý Y ¥ ` ¦¡§ T í £¦ C ' C e E ç¦ Â¡ E cåæ¡ ' ' É Ê ÷ ø G y ¡ ¶ ¡ ¤ ©¡¡¡ y ½ s ' é ¡ ¨ ÷ ø · T c 6 ½¾Â F e ' ST Á e ' S d ¡ ¦¡ d v # ¡¡2&')({| u¡
(ii) Ex 5: {(x, y ) : x2 + y 2 < 1} {(x, y ) : x2 + y 2 < 1} {(x, y ) : x2 + y 2 ≤ 1} [0, 1) [0, 1]
r ∈[0,1]
f (x) = x2 , X = [−1, 1], Y = [0, 1], A = [0, 1]. {(x, y ) : x2 + y 2 ≤ 1}
n=1
An ∼ [0, 1]∞ .
An
E
∞
ww ¿À ' · T S Á¿À C õ d WÃX ÃÄ T WX à « Å Æ ÇÈ ' WXÉÊ UV Å« ! "#ËÌ"Í$%')({|12 t vw # 8 u#2v
t0 ∈ E , E tn , tn → t 0 , tn ∈ E xn ∈ F d(tn , xn ) = r. xn N xn xk 0 , d(tn , xk0 ) = r, (n ≥ N ) d(t0 , xk0 ) = r, t0 ∈ E . xn d(xn , t0 ) ≤ d(xn , tn )+ d(tn , t0 ) = r + d(tn , t0 ), tn → t 0 , xn Bolzano-Weierstrass xn xn → x 0 , F x0 ∈ F , d(t0 , x0 ) = r. ( ) d(x0 , t0 ) ≤ d(x0 , xn ) + d(xn , tn ) + d(tn , t0 ) d(x0 , t0 ) ≥ d(xn , tn ) − d(x0 , xn ) − d(tn , t0 ). S (x, r) F ⊂ Rn r > 0, S (x, r) x r
x∈F
x0 ∈ E c , ε>0 f (x0 + ε) − f (x0 − ε) = 0, ( f (x0 −ε, x0 +ε) ) ∀y ∈ (x0 −ε, x0 +ε), δ = min{y −(x0 −ε), (x0 +ε)−y )}, f (y + δ ) − f (y − δ ) = 0, (x0 − ε, x0 + ε) ⊂ E c , x0 Ec Ec E
Ex 1.
{fj (x)}
¢¡¢£ ¤¢¥¢¦¢§¢¨¢¤¢¦© ! "#$%&')(012
Rn {x : fj (x) ≥ 1 }, (j, k = 1, 2, · · ·) k x : lim fj (x) > 0 .
j →∞
34 12 56
x : lim fj (x) > 0
[0, 1]
"#no$%' 7p!q % %2v }
∞ k=1
M,
rs t
[0, 1]
uvw8
|f (x1 ) + f (x2 ) + · · · + f (xn )| ≤ M,
E = {x ∈ [0, 1] : f (x) = 0}
C EE l g c v 2 v u #12' 7 u # vw 1# 8 %'({| } % ~ 6 G GGG c GG ' G T d¥¦§ ' WX¨© ~ ld ª d S¥ d ')« d '¢¡ £X¤ ' S « ¬GwG c °± ®¯ v 9 ({| p! rs ~ 6 ² c³´µ T
x0
r S (x0 , r) ∩ E ⊂
C (x1 , r ),
C (x1 , r )
x1
r,S (x0 , r) ∩ E r
r ∈Q+
x1 ∈ S (x0 , r) ∩ E ,
C (x1 , r ) ∩ E ⊂ E
C (x1 , r ) ∩ C (x2 , r ), x2 ∈ C (x1 , r ) ∩ E ,
F E E ⊂ F ⊂ F, E ∩ F = E = ∅. E, E ∩ F = ∅, F x1 ∈ F , O(x1 , 1) F c x2 ∈ O(x1 , 1) ∩ F , O(x1 , max{2, d(x1 , x2 )}), F x1 , x2 , · · ·, E = {x1 , x2 , · · ·}, E = ∅, E ∩ F = ∅, F x0 ∈ F , F xn xn → x 0 , E = {xn : n = 1, 2, · · ·}, E ∩F =∅ x0 ∈ F , F