※ 引述《ulin (狂傲魔羯)》之銘言：
> (i)【exterior measure】:
> d 00 00
> E is any subset of R , m*(E)=infΣ│Q│,where E包含於∪ Q
> j=1 j j=1 j
> Q is closed cube.
> 【Lebesque measure】:
> Given ε> 0,there exist an open set O with E包含於 O
> and m*(OE)≦ε
> 【σalgebra 】:
> d
> The set is a collection of subset of R ,that is closed under
> countable unions,countable intersections and complements.
> (ii)prove measurable set is σalgebra
> 【union】:let E = ∪ E ,where E is measurable,we want to prove
> j=1 j j
> E is measurable.
> <pf>: Given ε> 0,we may choose for each j an open set Q
> j 00 j
> with E 包含於Q & m*(Q  E )≦ε/2. Then O = ∪ Q is open,
> j j j j j=1 j
> 00 00
> and E包含於O, ∴m*(OE)≦m*[∪ (E Q)]≦Σm*(EQ)≦ε
> j=1 j j j=1 j j
> ∴ E is measurable.
> c
> 【complemrnt】:If E is measurable , then E is measurable.
> <pf>:E is measurable,then given any integer n , there exist an
> On with E包含於On and m(OnE)≦1/n,for n=1,2,3....
> c
> On is open,then On is closed and measurable.
> 00 c
> Let S = ∪ On is measurable.
> n=1
> c c 00 c c
> ∵E包含於On for all n ∴E 包含On ∴∪On包含於E
> n=1
> c 00 c
> and E ∪On包含於OE
> n=1
> c
> m*(E S)≦m*(OnE)≦1/n for all n
> c c
> => m*(ES)=0 => ES is measurable
> c c c c
> E =(ES)∪S (∵S包含E ) ∴ E is measurable.
> 【intersrction】:
> 00 00 c c
> E= ∩ E = (∪ E ) , E is measurable.
> j=1 j j=1 j
> 
> 【期中考第二題第三小題補齊】
(iii)There exists a closed set F with F包含於E and m(EF)≦ε,where
E is measurable.
(pf) c
E is measurable => E is measurable
c c
so there exist an open set O with E 包含於O and m(OE )≦ε
c c
let F=O is closed and F包含於E, and EF= OE
c
hence m(EF)=m(OE )≦ε

※發信站 [中央數學 織夢天堂 bbs.math.ncu.edu.tw]
•FROM [ulin 從 61222.dorm.ncu.edu.tw 發表]
