[文献] ・黒崎『集合論演習』第5章I(1);II(1)(4) ・伊藤『ルベーグ積分入門』p.7 ・志賀『集合への30講』第12講(p.75;183) ・野田宮岡『数理統計学の基礎』3.1.3-4(p.74) ・鈴木山田『数理統計学』p.6 ・盛田『実解析と測度論の基礎』1.3面積と積分(p.25) ・『岩波数学辞典』225測度論C(p.627) |
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[関連事項] 数列の上極限/関数の上極限 |
集合列{An}の上極限集合superior limitないし最大極限集合 |
|
とは、 | |
|
|
∞ | ∞ |
=( | ∞ | )∩( | ∞ | )∩… | |||||
∩ |
∪ |
Am |
∪ | Am | ∪ | Am | |||||
n=1 | m=n |
m=1 | m=2 |
lim sup |
An | ないし |
lim |
An |
||||
n→∞ | n→∞ |
∞ | ∞ |
=( | ∞ | )∩( | ∞ | )∩… | |||||||||
ω∈ | lim sup | An = |
∩ |
∪ |
Am |
∪ | Am | ∪ | Am | ||||||
n→∞ | n=1 | m=n |
m=1 | m=2 |
∞ |
=(An∪An+1∪An+2∪An+3∪…) | |||||
命題2: | 任意の自然数nに対して、ω∈ | ∪ |
Am |
が満たされる。 |
||
m=n |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
lim inf |
An | ないし |
lim |
An |
||||
n→∞ | n→∞ |
∞ | ∞ |
=( | ∞ | )∪( | ∞ | )∪… | |||||
∪ | ∩ | Am |
∩ | Am | ∩ | Am | |||||
n=1 | m=n |
m=1 | m=2 |
lim inf |
An | ないし |
lim |
An |
||||
n→∞ | n→∞ |
∞ | ∞ |
=( | ∞ | )∪( | ∞ | )∪… | |||||||||
ω∈ | lim inf |
An = |
∪ | ∩ | Am |
∩ | Am | ∩ | Am | ||||||
n→∞ | n=1 | m=n |
m=1 | m=2 |
∞ |
=(An∩An+1∩An+2∩An+3∩…) | |||||
命題2: | (∃n∈N) (∀m∈N) (m≧n ⇒ ω∈ | ∩ | Am |
が満たされる。 |
||
m=n |
∞ |
|||
ω∈ | ∩ | Am |
|
m=n |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
lim sup | An |
⊃ |
lim inf | An | ||||
n→∞ | n→∞ |
lim | An ⊃ | lim |
An | |||||
n→∞ | n→∞ |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
|
( | lim sup | An |
) | c |
=
|
lim inf | An c | ||||
n→∞ | n→∞ |
( | lim | An |
) | c | = | lim |
An c | |||||
n→∞ | n→∞ |
( | lim inf | An |
) | c |
=
|
lim sup | An c | ||||
n→∞ | n→∞ |
( | lim | An |
) | c | = |
lim |
An c | |||||
n→∞ | n→∞ |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
|
lim | An |
||||
n→∞ |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
[野田宮岡『数理統計学の基礎』3.1.2(p. 73);鈴木山田『数理統計学』p.6;
盛田『実解析と測度論の基礎』1.3面積と積分(p.25);]
集合列{An}が
An⊂ An+1 (n=1,2,…)
を、満たすとき、「増大列である」「増加列である」という。
→[トピック一覧:極限集合] → 集合論目次・総目次 |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
|
→[トピック一覧:極限集合] → 集合論目次・総目次 |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
増大列では極限 |
lim | An |
が存在し、 |
|||||
n→∞ |
∞ |
||||
lim An = | ∪ |
Ak |
||
n→∞ | k=1 |
lim sup |
An = | lim |
An | ||||
n→∞ | n→∞ |
∞ | ∞ |
=( | ∞ | )∩( | ∞ | )∩… | |||||
= | ∩ |
∪ |
Am |
∪ | Am | ∪ | Am | ||||
n=1 | m=n |
m=1 | m=2 |
=( | ∞ | )∩( | ∞ | )∩… | |||||
|
∪ | Am | ∪ | Am | |||||
m=1 | m=1 |
∞ | = |
∞ | =…= | ∞ | = | ∞ | =… |
||||||
∵An⊂ An+1(n=1,2,…)より、 | ∪ | Am | ∪ | Am | ∪ | Am | ∪ | Am | |||||
m=1 | m=2 | m=n | m=n+1 |
=( |
∞ | ) | ||
∪ | Am | |||
m=1 |
lim inf |
An = | lim | An | |
n→∞ | n→∞ |
∞ | ∞ |
( | ∞ | )∪( | ∞ | )∪… | |||||||
= | ∪ | ∩ | Am |
= | ∩ | Am | ∩ | Am | |||||
n=1 | m=n |
m=1 | m=2 |
=( |
∞ | ) | ||
∪ | An | |||
n=1 |
したがって、増大列{An }では、 | lim | An = | lim |
An | が成り立ち、 | lim |
An | が存在する。 | |||||
n→∞ | n→∞ | n→∞ |
∞ |
||||
lim An = | ∪ |
Ak |
||
n→∞ | k=1 |
∞ | ∞ |
∞ | ∞ | ||||||||||||||
lim sup |
An = | lim |
An | = |
∩ | ∪ |
Am = ( | ∪ | Am | )∩( | ∪ |
Am | )∩… |
||||
n→∞ | n→∞ | n=1 | m=n | m=1 | m=2 |
∞ | |||
∪ | Am | =(−∞,1]∪(−∞,2]∪…∪(−∞,100]∪…∪(−∞,1000]∪…∪(−∞,∞)=R | |
m=1 |
∞ | |||
∪ | Am | = (−∞,2]∪…∪(−∞,100]∪…∪(−∞,1000]∪…∪(−∞,∞)=R | |
m=2 |
∞ | |||
∪ | Am | = (−∞,100]∪…∪(−∞,1000]∪…∪(−∞,∞)=R | |
m=100 |
∞ | |||
∪ | Am | = (−∞,1000]∪…∪(−∞,∞)=R | |
m=1000 |
lim sup |
An = | lim |
An | = |
R∩R∩…=R | ||||
n→∞ | n→∞ |
∞ | ∞ |
∞ | ∞ |
|
|||||||||||
lim inf |
An = | lim | An | = |
∪ | ∩ | Am = ( | ∩ | Am | )∪( | ∩ | Am | )∪… |
||
n→∞ | n→∞ | n=1 | m=n | m=1 | m=2 |
∞ | |||
∩ | Am | =(−∞,1]∩(−∞,2]∩…∩(−∞,100]∩…∩(−∞,1000]∩…=(−∞,1] | |
m=1 |
∞ | |||
∩ | Am | = (−∞,2]∩…∩(−∞,100]∩…∩(−∞,1000]∩…=(−∞,2] | |
m=2 |
∞ | |||
∩ | Am | = (−∞,100]∩…∩(−∞,1000]∩…=(−∞,100] | |
m=100 |
∞ | |||
∩ | Am | = (−∞,1000]∩…=(−∞,1000] | |
m=1000 |
∞ | ||||
lim |
∩ | Am | =(−∞,∞)=R | |
m→∞ |
m |
lim inf |
An = | lim | An | = |
(−∞,1]∪(−∞,2]∪…∪(−∞,100]∪…∪(−∞,1000]∪…∪(−∞,∞) = (−∞,∞) = R | |
||
n→∞ | n→∞ |
したがって、確かに、 | lim | An = | lim |
An | が成り立ち、 | lim |
An | が存在する。 | |||||
n→∞ | n→∞ | n→∞ |
∞ |
||||
lim An = | (−∞,∞) = R = | ∪ |
Ak |
|
n→∞ | k=1 |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
|
減少列では極限 |
lim | An |
が存在し、 |
|||||
n→∞ |
∞ |
||||
lim An = | ∩ | Ak |
||
n→∞ | k=1 |
lim sup |
An = | lim |
An | ||||
n→∞ | n→∞ |
∞ | ∞ |
=( | ∞ | )∩( | ∞ | )∩… | |||||
= | ∩ |
∪ |
Am |
∪ | Am | ∪ | Am | ||||
n=1 | m=n |
m=1 | m=2 |
∞ | =Ak | ||
∵An⊃An+1(n=1,2,…)より、 | ∪ | Am | |
m=k |
= |
∞ | ||
∩ | Ak | ||
k=1 |
lim inf |
An = | lim | An | |
n→∞ | n→∞ |
∞ | ∞ |
( | ∞ | )∪( | ∞ | )∪… | |||||||
= | ∪ | ∩ | Am |
= | ∩ | Am | ∩ | Am | |||||
n=1 | m=n |
m=1 | m=2 |
( | ∞ | )∪( | ∞ | )∪… | |||||
= | ∩ | Am | ∩ | Am | |||||
m=1 | m=1 |
∞ | = |
∞ | =… | ||||
∵An⊃An+1(n=1,2,…)より、 | ∩ | Am | ∩ | Am | |||
m=1 | m=2 |
=( |
∞ | ) | ||
∩ | Ak | |||
k=1 |
したがって、減少列{An }では、 | lim | An = | lim |
An | が成り立ち、 | lim |
An | が存在する。 | |||||
n→∞ | n→∞ | n→∞ |
∞ |
||||
lim An = | ∩ | Ak |
||
n→∞ | k=1 |
∞ | ∞ |
∞ | ∞ | ||||||||||||||
lim sup |
An = | lim |
An | = |
∩ | ∪ |
Am = ( | ∪ | Am | )∩( | ∪ |
Am | )∩… |
||||
n→∞ | n→∞ | n=1 | m=n | m=1 | m=2 |
∞ | |||
∪ | Am | =(−∞,−1]∪(−∞,−2]∪…∪(−∞,−100]∪…∪(−∞,−1000]∪…∪(−∞,−∞)= (−∞,−1] = A1 | |
m=1 |
∞ | |||
∪ | Am | = (−∞,−2]∪…∪(−∞,−100]∪…∪(−∞,−1000]∪…∪(−∞,−∞)= (−∞,−2] = A2 | |
m=2 |
∞ | |||
∪ | Am | = (−∞,−100]∪…∪(−∞,−1000]∪…∪(−∞,−∞)=(−∞,−100]= A100 |
|
m=100 |
∞ | |||
∪ | Am | = (−∞,−1000]∪…∪(−∞,−∞)=(−∞,−1000]= A1000 |
|
m=1000 |
∞ | ||||
lim |
∪ | Am | =(−∞,−∞)=φ |
|
m→∞ |
m |
lim sup |
An = | lim |
An | = |
A1∩A2∩…∩A100∩…∩A1000∩…=(−∞,−1]∩(−∞,−2]∩…∩(−∞,−100]∩…∩(−∞,−1000]∩…=φ | ||||
n→∞ | n→∞ |
∞ | ∞ |
∞ | ∞ |
|
|||||||||||
lim inf |
An = | lim | An | = |
∪ | ∩ | Am = ( | ∩ | Am | )∪( | ∩ | Am | )∪… |
||
n→∞ | n→∞ | n=1 | m=n | m=1 | m=2 |
∞ | |||
∩ | Am | =(−∞,−1]∩(−∞,−2]∩…∩(−∞,−100]∩…∩(−∞,−1000]∩…∩(−∞,−∞)=φ | |
m=1 |
∞ | |||
∩ | Am | = (−∞,−2]∩…∩(−∞,−100]∩…∩(−∞,−1000]∩…∩(−∞,−∞)=φ | |
m=2 |
∞ | |||
∩ | Am | = (−∞,−100]∩…∩(−∞,−1000]∩…∩(−∞,−∞)=φ | |
m=100 |
∞ | |||
∩ | Am | = (−∞,−1000]∩…∩(−∞,−∞)=φ | |
m=1000 |
lim inf |
An = | lim | An | = |
φ∪φ∪…∪φ∪…∪φφ∪…∪φ=φ | ||
n→∞ | n→∞ |
したがって、確かに、 | lim | An = | lim |
An | が成り立ち、 | lim |
An | が存在する。 | |||||
n→∞ | n→∞ | n→∞ |
∞ |
|||||
lim An = | (−∞,−∞) = φ = |
|
Ak |
||
n→∞ | k=1 |
→[トピック一覧:極限集合] → 集合論目次・総目次 |
|
→[トピック一覧:極限集合] → 集合論目次・総目次 |