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作者  b218h (Gordon @ UBC) 站內  97RA_I
標題  [問題]
時間  2008/10/15 Wed 12:45:51


Lemma: 

    Fix a Lebesgue-Stieljes measure μ , and call it μ and denote its domain
                                       F

by Μ (i.e. the sigma-algebra of all μ*-measurable sets)
                                        F

Then,
             μ(E)=sup{ μ(C): C C E, C is compact}, for any E in Μ
                                   ̄


Problem:

   Show that E is in Μ if and only if E=H∪N, where H is F  , μ(N)=0
                                                            δ

[pf]

    "<=" trivial.

    "=>"
         Let E be in Μ, and write

                               E=  ∪ E∩[-k,k]
                                   k≧1

         By the lemma, for each k in |N, j in |N, ε>0,there exists a corres-

         ponding compact set
                                 C    C E∩[-k,k]
                                  k,j  ̄

                        ┌    ┐    ┌          ┐     1    1
                      μ│C   │>μ│E∩[-k,k]│- ── ── ε
                        └ k,j┘    └          ┘    2^j  2^k

         Define
                            ┌        ┐
                       H=∪│ ∩ C   │ => H is F  and H C E
                           k└  j  k,j┘          δ       ̄


         Define
                                        N=E╲H


         Note that N is measurable, and therefore N is in Μ, and

         For all ε>0,
                              ┌                                 ┐
                              │                   ┌          ┐│
           μ(N)=μ(E╲H)=μ│∪ E∩[-k,k] ╲ ∪│ ∩ C     ││
                              │ k                s└  j  s,j  ┘│
                              └                                 ┘
                                    ┌                              ┐
                              ∞    │                ┌          ┐│
                          ≦  Σ  μ│E∩[-k,k] ╲ ∪│ ∩ C     ││
                             k=1   │               s└  j  s,j  ┘│
                                    └                              ┘

                              ∞    ┌                      ┐
                          ≦  Σ  μ│E∩[-k,k] ╲ ∩ C    │
                             k=1   └               j  k,j ┘


                              ∞   ∞    ┌                 ┐
                          =  Σ   Σ  μ│E∩[-k,k] ╲C   │
                             k=1 j=1   └              k,j┘

                              ∞   ∞    1    1
                          ≦  Σ   Σ  ── ── ε
                             k=1 j=1  2^j  2^k

                          = ε

By taking ε─→0 , we have

                                μ(N)=0


--
這樣 ok 嗎^^? 一直改來改去的...

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□ Modify: 2008/10/16 Thu 01:13:59  128.189.230.119 修改

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