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作者  fongya (......) 站內  97RA_I
標題  CH.2 Exercise 11
時間  2008/12/07 Sun 22:39:16

972201011

 11.Prove that
                           d
    if f is integrable on R  ,real-valued,

    and ∫  f(x)dx ≧ 0 for every measurable E,
          E

    then f(x) ≧ 0  a.e.x .


    As a result ∫  f(x)dx = 0  for every measurable E,
                 E

    then f(x) = 0  a.e.




Proof.

    (1)

       For all k belongs to N,

       Let E  = {f(x)<-1/k}
            k

       => E  is measurable
           k

           and 0 ≦ ∫  f(x)dx ≦ -1/k m (E ) ≦ 0
                     E_k                   k


       => m(E ) = 0
             k
                                ∞          ∞
       Hence m({f(x)<0}) = m ( ∪  E  ) ≦ Σ  m( E ) = 0
                                k=1  k      k=1     k

    (2)
       By (1) , we also have ∫f(x)dx ≦ 0
                              E
                for all measurable E

       => f(x)dx ≦ 0  a.e.x


       Hence ∫f(x)dx = 0 , for all measurable E
              E

       f(x)≧0 and f(x)≦0  a.e. x

       => f(x)=0  a.e. x


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□ Modify: 2008/12/07 Sun 22:55:17  114-43-64-129.dynamic.hinet.net 修改

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