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作者  jksmay (哈姆) 站內  97RA_I
標題  the conditions of B.C.T. (thm 1.4) are sharp
時間  2008/11/17 Mon 12:39:47

972201013

Question:

Can we release the conditios on the B.C.T. (thm 1.4) ?


Answer:

  the answer is "no".

  that is, the condictions of B.C.T. are sharp

(i) f_n's are not bounded

    for positive integer n,

    let f_n be defined on [0,1] by

    f_n(x)={ n, if x is in [ 0,1/n ]
           {
           { 0, if x is in ( 1/n,1 ]

    then {f_n} is a sequence of Lebesgue integrable with unbounded

    function on [0,1]

    for 0<x≦1, we have f_n(x)→f(x)=0   as n→∞

      where f(x)={ 0, if x is in (0,1]
                 {
                 {∞, is x=0

    since ∫ f_n(x)dx = n * m([ 0,1/n ])+ 0 * m(( 1/n,1 ]) =1 for all n
         [0,1]

    therefore,  ∫ f_n(x)dx 不趨近於 ∫ f(x)dx =0  as  n→∞
               [0,1]                [0,1]


(ii)supp(f_n) contains in E , with m(E) = ∞

    for positive integer n,

    let f_n be defined on [0,∞) by

    f_n(x)={ 1/n, if x is in [ 0,n ]
           {
           { 0  , if x > n

    for every x ≧0, we have f_n(x)→f(x)=0   as n→∞

    since ∫ f_n(x)dx = 1 , for all n
         [0,∞)

          supp(f_n)=[0,n] →[0,∞)  as n→∞

    therefore , we have

    ∫ f_n(x)dx  不趨近於 ∫ f(x)dx = 0  as n→∞
   [0,∞)                [0,∞)


by the above two examples, we know that the conditions of

Bounded Convergence Theorem are sharp.

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