测度与积分

m∗ (Br (x) \ Ek ) m(Br (x))
= 0. §-
m8V ⊃ Ek §¦ m(V ) < +∞" δ =
2k+1 m(V )
B = {Br (x) : Br (x) ⊂ V, x ∈ Ek , m∗ (Br (x)) ≤ δm(Br (x))}. B ´fine cover§ ¤ ± • 3 üü Ø "¤± 2k+2
§fk ∈ Lp (D)§…3Lp (D)¥Âñ"f (r cos θ, r sin θ)dθ, r ∈ (0, 1).
0
1) p = 1ž§¦yé?¿ a ∈ (0, 1)§ϕk 3L1 ((a, 1)) þÂñ" 2) p > 2ž§¦yϕk 3Lp ((0, 1))þÂñ" ———————————————————————–
= = =
− y )g (y )dydx = R R f (x − y )g (y )dxdy R g (y ) R f (x − y )dxdy R g (y ) R f (x)dxdy = R g (y )dy R f (x)dx.
R R f (x
...........................5
|ϕk (r) − ϕm
Lp (D)
(r)|p rdθdr
1 p
1 2π 0 0
1 r
p p
drdθ
≤ C ϕk − ϕm ¤±
k,m→+∞
lim
ϕk − ϕm
L1 ((0,1))
=0
..................4© ———————————————————————–
l E ⊂ Rn "é?¿x ∈ E §k lim
Ón§eA2 = {(x, y ) : lim sup fk (x, y ) = lim inf gk (x, y )},
k→+∞ k→+∞
A2 Œÿ"KA1 ∩ A2 = {lim fk , lim gk ∃} Œÿ"................1© B = {x ∈ A1 ∩ A2 : limk→+∞ fk = limk→+∞ gk } = R2 \ ∪a,b∈Q,a>b ({ lim fk > a} ∩ {b > lim gk })
m(∪∞ i=1 Ui ) =
∞ i=1 m(Vi )
= limk→+∞
k i=1 m(Vi )
= limk→+∞ m(∪k i=1 Vi ) = limk→+∞ m(Ui )"
-k → +∞"............2 ———————————————————————– fk (x, y )§gk (x, y ) ´R2 þŒÿ¼ê"®•éa.e.
E
f §?¿
1) ¦yµfk §f 3E þŒÈ" 2) ¦yµ lim
k→+∞ X
¤±
k,m→+∞
ϕk − ϕm
L1 ((a,1))
=0
..................6©
2π 0( 0 2π 0( 0
0 |ϕk (r )
− ϕm |dr = ≤ ≤
|ϕk (r) − ϕk (r)|dθ)dr |ϕk (r) − ϕm (r)| r 1 dθdr
rp
1 p 1 p
1 2π 0 0
∞ i=1 |f (xi )|
< +∞§¤±k
∞ p i=1 |f (xi )|
< +∞".........1©
o E ´Rn þŒÿ8"é?¿ ϕ ∈ D(Rn )§ ϕdx = 0.
E
¦yµm(E ) = 0" -F = χE (x)" ϕk ∈ D(Rn )§ ¦ m(E ) = 0".............5© F − ϕk
L∞ (E )
= 2§•3š0ÿ8E1 §f |E1 ≥ 2 − "
¤±f − 1|E1 ≥ 1 − "..........................5© ———————————————————————– 8 f, g ∈ L1 (R)"-ϕ(x) = f 1) ϕ ∈ L1 (R)§…k ϕ 2) f ∈ g (x) =
2k
2)é ? ¿ k § Œ ÿ 8Uk § ¦ m(Uk \ E ) < 1 k §∀k "¤±U \ E Œÿ§¤±E Œÿ"
1 k " -U
= ∩Uk " Km(U \ E ) ≤
———————————————————————– Ê y²µE ´Rn ¥ÿÝk• Œÿ8§fk 3E þŒÿ"XJfk 3E þa.e.Âñ > 0§•3F ⊂ E §¦ m(E \ F ) < §…fk 3F þ˜—Âñ f " › E ⊂ Rn ´Œÿ8§µ(E ) < +∞"fk ´E þŒÿ¼ê§a.e. Âñ f "®• |fk |(1 + log |fk |)dx < Λ < +∞.
m
B1 §· · · §Bm ∈ B § ¦
m
m∗ (E \
m i=1 Bi )
<
m∗ (∪m i=1 Bi \ Ek ) =
i=1
m∗ (Bi \ Ek ) ≤ δ
i=1
m∗ (Bi ) ≤ δm(V ) =
2k+1
. "
∗ m8V ⊃ ∪m i=1 (Bi \Ek )§¦ m(V ) ≤ 2k+1 "-Vk = V ∪V "Km (Vk \Ek ) < ∗ -U = ∪i Vi §m (U \ E ) ≤ k m(Vk \ Ek ) < "
r→0
m∗ (Br (x) \ E ) m(Br (x))
= 0.
¦yµ 1) ?¿ 2) E Œÿ ———————————————————————– -Ek = E ∩ Bk (0)"Km∗ (Ek ) < +∞"w,é?¿ x ∈ Ek §E,k lim
r →0
§•3Œÿ8U ⊃ E §¦ m∗ (U \ E ) <
R
m(Ey )dy
..............4© d®•§m(Ey ) = 0"..............1© ———————————————————————– n X ´˜‡Ã•8"?¿E ⊂ X §½Â E ´Ã•8 +∞ 0 E=∅ µ(E ) = E ¥ ƒ ‡ê Ù§ yµ 1). µ´ ýÝ" 2). X ?¿f8ÜŒÿ" 3). f ∈ Lp (X )§p ≥ 1§Kf 3–õŒ ‡:ƒ Ñ•0" 4). p > 1§f ∈ Lp (X )žkf ∈ L1 (X )" ———————————————————————– µ∗ (∅) = 0
i m(Ai )
= +∞§k µ∗ (∪i Ai ) ≤
i
µ∗ (Ai ).
iµ
∗ (A
i)
< +∞ž§Ai ¥•kk•‡Ø´˜8" m(Ai ) = ∞" A∩T = m(Ai ) ..................2©
?¿ A1 §· · · §Ai §· · · §XJAi ¥kÕ8§Km(∪i Ai ) = XJÑ´k•8§Kw,km(∪i Ai ) = X §µ∗ (T )
2)
suppϕ ⊂ BR "K3BR+1+|x| þ•3L | f (x − y + h) − f (x − y ) | ≤ L, ∀|h| < 1. h | f (x − y + h) − f (x − y ) ||g | ≤ L|g |. h
f (x−y +h)−f (x−y ) gdy h
K
∞ i→+∞
lim µ(Ui ) = µ(
i=1
Ui ).
———————————————————————– m(Ui ) ≤ m(Ui+1 ) ≤ m(∪i Ui )§¤±limi→+∞ m(Ui )k¿Â"
k ∞ ∞ -V1 = U1 §V2 = U1 \ U1 §· · · "·‚k∪k i=1 Vi = ∪i=1 Ui §∪i=1 Vi = ∪i=1 Ui "............3
k→+∞ k→+∞
y ∈ R1 k
lim fk (x, y ) = lim gk (x, y ), a.e.x ∈ R.
¦yµéa.e.
(x, y ) ∈ R2 §k
k→+∞
lim fk (x, y ) = lim gk (x, y ).
k→+∞
———————————————————————– A1 = {(x, y ) : lim sup fk (x, y ) = lim inf gk (x, y )}
d››Âñ½n§ limh→0 .......................5 ———————————————————————– Ô -D´R2 ¥± :• % ü ϕk (r) = 1 2π
2π R f (x−y +h)−f (x−y ) g (y )dy h
= =
Leabharlann Baidu
Rf
R limh→0
(x − y )g (y )dy
˜uŒÆ
•Á‘§µ¢C¼ê
6¶µ
‰)•ÁÁK;^’
žmµ c 06
ÆÒµ
24 F
(Aò)
•?µ
`² 1. g•Kò ›˜K§Ù¥1›˜K´À‰K"
2. •Ážm•3‡ ž" 3. XJ؉AÏ`²§Rn þ ÿÝÑ´• Lebesgue ÿÝ"
˜ (X, M, µ)´ÿݘm§Ui ⊂ M§…Ui ⊂ Ui+1 §i = 1, 2, 3, · · · "¦yµ
k→+∞ k→+∞
Ï•lim supk→+∞ §lim inf k→+∞ Œÿ§ A1 = R2 \ ∪a,b∈Q,a>b {lim supk→+∞ fk > a > b > lim inf k→+∞ fk } = R2 \ ∪a,b∈Q,a>b ({{lim supk→+∞ fk > a} ∩ {b > lim inf k→+∞ fk }). ¤±A1 Œÿ".................2©
1 a |ϕk (r )
− ϕm |dr = ≤ ≤ = ≤ lim
2π 1 a ( 0 |ϕk (r ) − ϕk (r )|dθ )dr 2π 1 dr a ( 0 |ϕk (r ) − ϕm (r )|rdθ a 2π 1 dr 0 ( 0 |ϕk (r ) − ϕm (r )|dθ )r a 1 2π dr 0 0 |ϕk (r ) − ϕm (r )|rdθ a 1 a D |ϕk − ϕm |
R f (x
− y )g (y )dy "¦yµ
L1 (R) ≤ f L1 (R) g L1 (R) " 1 C (R)§g ´;|¼êž§ϕ::Œ‡§…ϕ =
f
g"
———————————————————————– f (x − y )ϕ(y )3R2 þŒÿ§dTonelli½n§ϕŒÿ"
R ϕ(x)dx
L1 (Rn )
→ 0" K F − ϕk
L1 (E )
→ 0§
Ê E ´Rn ¥Œÿ8§0 ≤ f ≤ 2§a.e.x ∈ E " f ¦yµ f (x) − 1
L∞ (E ) L∞ (E )
= 2. = 1.
———————————————————————– ®•0 ≤ f ≤ 2§K−1 ≤ f − 1 ≤ 1§¤±|f − 1| ≤ 1" f
-f 3x1 §· · · §xi §· · · Ñ´0"-fk = | ´4O {ü¼ê "¤± |f |p dµ = lim
X
k p i=1 f (xi )χ{xi } |
k k→+∞ X
∞
|f (xi )
i=1
χ{xi } |p =
i=1
|f (xi )|p .
.............2© ¤±f ∈ L1 Ñ
...............2©
1 -Ek = {x ∈ X : |f | ≥ k }" K k1p µ(Ek ) ≤ ±{x : f > 0}´Œê "...........3© X
|f |p dµ < +∞" ¤ ±Ek ´ k • 8 § ¤ =
k p i=1 |f (xi )| χ{xi } §fk
A ⊂ X"é?¿ T ⊂ < +∞" T = {x1 , · · · , xk }" Ø ” {x1 , · · · , xi }§KT \ A = {xi+1 , · · · , xk }§Kk k = µ∗ (T ) = i + (k − i) = µ∗ (T ∩ A) + µ∗ (T \ A).
k→+∞ k→+∞
= R2 \ ∪a,b∈Q,a>b ({ lim gk > a} ∩ {b > lim fk })
k→+∞ k→+∞
B ´Œÿ8"....................2© -E = R2 \ B §Ey = {x : (x, y ) ∈ E }..............1© dFubini½n§·‚k m(E ) =