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作者  ulin (狂傲魔羯) 站內  97RA_I
標題  Re: 期中考第二題
時間  2008/11/11 Tue 10:05:11

※ 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*(O-E)≦ε
>   【σ-algebra 】:
>                                          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 σ-algebra
>    【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*(O-E)≦m*[∪ (E- Q)]≦Σm*(E-Q)≦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(On-E)≦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O-E
>                  n=1
>               c
>           m*(E -S)≦m*(On-E)≦1/n for all n
>                 c          c
>           => m*(E-S)=0 =>  E-S is measurable
>            c   c               c       c
>           E =(E-S)∪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(E-F)≦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(O-E )≦ε
             c                                    c
      let F=O is closed and F包含於E, and E-F= O-E
                        c
      hence m(E-F)=m(O-E )≦ε

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