Prove Lusin's Theorem with characteristic function and simple function.











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Lusin's Theorem: Suppose $f$ is a complex measurable on $X$, $mu(A)<infty$, $f(x)=0$ if $xnotin A$ and $epsilon >0$. Then there exists a $g in C_c(X)$ such that $mu ({x:f(x)neq g(x)}) <epsilon$.



In Rudin page 55-56, author proves with the general case of $f$.
Here, I want to prove theorem with $f$ is a characteristic function, $f$ is a simple function because Rudin's proof is long and complex.



Please hint me!
Thank you!






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    up vote
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    favorite












    Lusin's Theorem: Suppose $f$ is a complex measurable on $X$, $mu(A)<infty$, $f(x)=0$ if $xnotin A$ and $epsilon >0$. Then there exists a $g in C_c(X)$ such that $mu ({x:f(x)neq g(x)}) <epsilon$.



    In Rudin page 55-56, author proves with the general case of $f$.
    Here, I want to prove theorem with $f$ is a characteristic function, $f$ is a simple function because Rudin's proof is long and complex.



    Please hint me!
    Thank you!






    measure-theory







    share|cite|improve this question


























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Lusin's Theorem: Suppose $f$ is a complex measurable on $X$, $mu(A)<infty$, $f(x)=0$ if $xnotin A$ and $epsilon >0$. Then there exists a $g in C_c(X)$ such that $mu ({x:f(x)neq g(x)}) <epsilon$.



      In Rudin page 55-56, author proves with the general case of $f$.
      Here, I want to prove theorem with $f$ is a characteristic function, $f$ is a simple function because Rudin's proof is long and complex.



      Please hint me!
      Thank you!






      measure-theory







      share|cite|improve this question















      Lusin's Theorem: Suppose $f$ is a complex measurable on $X$, $mu(A)<infty$, $f(x)=0$ if $xnotin A$ and $epsilon >0$. Then there exists a $g in C_c(X)$ such that $mu ({x:f(x)neq g(x)}) <epsilon$.



      In Rudin page 55-56, author proves with the general case of $f$.
      Here, I want to prove theorem with $f$ is a characteristic function, $f$ is a simple function because Rudin's proof is long and complex.



      Please hint me!
      Thank you!






      measure-theory




      measure-theory






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      edited yesterday









      Davide Giraudo

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      asked Apr 11 '13 at 16:09









      Đặng Phước nhật

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