Any continuous function is Baire function?











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Afirmation. All continuous function is Baire function.



Let $f:Xto mathbb{R}$ continuous function. X compact hausdorff with $mathcal{B} sigma$-algebra of Baire.
Let $bigcap_{n=1}^{infty} O_ninmathcal{B}(mathbb{R})$ compact set. (sigma-algebra of Baire in $mathbb{R}$)
Now, $f^{-1}(bigcap_{n=1}^{infty} O_n)=bigcap_{n=1}^{infty}f^{-1}(O_n)$ compact set, and $f^{-1}(O_n)$ open sets. Therefore, $f^{-1}(bigcap_{n=1}^{infty} O_n)in mathcal{B}(X)$. Therefore $f$ is Baire function.



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    up vote
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    A have doubt.



    Afirmation. All continuous function is Baire function.



    Let $f:Xto mathbb{R}$ continuous function. X compact hausdorff with $mathcal{B} sigma$-algebra of Baire.
    Let $bigcap_{n=1}^{infty} O_ninmathcal{B}(mathbb{R})$ compact set. (sigma-algebra of Baire in $mathbb{R}$)
    Now, $f^{-1}(bigcap_{n=1}^{infty} O_n)=bigcap_{n=1}^{infty}f^{-1}(O_n)$ compact set, and $f^{-1}(O_n)$ open sets. Therefore, $f^{-1}(bigcap_{n=1}^{infty} O_n)in mathcal{B}(X)$. Therefore $f$ is Baire function.



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      A have doubt.



      Afirmation. All continuous function is Baire function.



      Let $f:Xto mathbb{R}$ continuous function. X compact hausdorff with $mathcal{B} sigma$-algebra of Baire.
      Let $bigcap_{n=1}^{infty} O_ninmathcal{B}(mathbb{R})$ compact set. (sigma-algebra of Baire in $mathbb{R}$)
      Now, $f^{-1}(bigcap_{n=1}^{infty} O_n)=bigcap_{n=1}^{infty}f^{-1}(O_n)$ compact set, and $f^{-1}(O_n)$ open sets. Therefore, $f^{-1}(bigcap_{n=1}^{infty} O_n)in mathcal{B}(X)$. Therefore $f$ is Baire function.



      It is correct?










      share|cite|improve this question















      A have doubt.



      Afirmation. All continuous function is Baire function.



      Let $f:Xto mathbb{R}$ continuous function. X compact hausdorff with $mathcal{B} sigma$-algebra of Baire.
      Let $bigcap_{n=1}^{infty} O_ninmathcal{B}(mathbb{R})$ compact set. (sigma-algebra of Baire in $mathbb{R}$)
      Now, $f^{-1}(bigcap_{n=1}^{infty} O_n)=bigcap_{n=1}^{infty}f^{-1}(O_n)$ compact set, and $f^{-1}(O_n)$ open sets. Therefore, $f^{-1}(bigcap_{n=1}^{infty} O_n)in mathcal{B}(X)$. Therefore $f$ is Baire function.



      It is correct?







      functional-analysis analysis measure-theory






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      edited Nov 18 at 16:40

























      asked Nov 18 at 16:33









      eraldcoil

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          The Baire-$sigma$-algebra is generated by closed $G_delta$-sets. If $bigcap_{i=1}^infty O_i$ is closed with open $O_n$, then also $f^{-1}(bigcap_{i=1}^infty O_i)$ is closed. Moreover $f^{-1}(bigcap_{i=1}^infty O_i) = bigcap_{i=1}^infty f^{-1}(O_i)$. Again using the continuity we see that $f^{-1}(O_i)$ is open. Thus $f^{-1}(bigcap_{i=1}^infty O_i)$ is Baire-measurable.



          Since we only need to prove measurability on a generating set-system, we get that $f$ is measurable according to the Baire-$sigma$-algebras.






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            The Baire-$sigma$-algebra is generated by closed $G_delta$-sets. If $bigcap_{i=1}^infty O_i$ is closed with open $O_n$, then also $f^{-1}(bigcap_{i=1}^infty O_i)$ is closed. Moreover $f^{-1}(bigcap_{i=1}^infty O_i) = bigcap_{i=1}^infty f^{-1}(O_i)$. Again using the continuity we see that $f^{-1}(O_i)$ is open. Thus $f^{-1}(bigcap_{i=1}^infty O_i)$ is Baire-measurable.



            Since we only need to prove measurability on a generating set-system, we get that $f$ is measurable according to the Baire-$sigma$-algebras.






            share|cite|improve this answer

























              up vote
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              The Baire-$sigma$-algebra is generated by closed $G_delta$-sets. If $bigcap_{i=1}^infty O_i$ is closed with open $O_n$, then also $f^{-1}(bigcap_{i=1}^infty O_i)$ is closed. Moreover $f^{-1}(bigcap_{i=1}^infty O_i) = bigcap_{i=1}^infty f^{-1}(O_i)$. Again using the continuity we see that $f^{-1}(O_i)$ is open. Thus $f^{-1}(bigcap_{i=1}^infty O_i)$ is Baire-measurable.



              Since we only need to prove measurability on a generating set-system, we get that $f$ is measurable according to the Baire-$sigma$-algebras.






              share|cite|improve this answer























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                up vote
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                The Baire-$sigma$-algebra is generated by closed $G_delta$-sets. If $bigcap_{i=1}^infty O_i$ is closed with open $O_n$, then also $f^{-1}(bigcap_{i=1}^infty O_i)$ is closed. Moreover $f^{-1}(bigcap_{i=1}^infty O_i) = bigcap_{i=1}^infty f^{-1}(O_i)$. Again using the continuity we see that $f^{-1}(O_i)$ is open. Thus $f^{-1}(bigcap_{i=1}^infty O_i)$ is Baire-measurable.



                Since we only need to prove measurability on a generating set-system, we get that $f$ is measurable according to the Baire-$sigma$-algebras.






                share|cite|improve this answer












                The Baire-$sigma$-algebra is generated by closed $G_delta$-sets. If $bigcap_{i=1}^infty O_i$ is closed with open $O_n$, then also $f^{-1}(bigcap_{i=1}^infty O_i)$ is closed. Moreover $f^{-1}(bigcap_{i=1}^infty O_i) = bigcap_{i=1}^infty f^{-1}(O_i)$. Again using the continuity we see that $f^{-1}(O_i)$ is open. Thus $f^{-1}(bigcap_{i=1}^infty O_i)$ is Baire-measurable.



                Since we only need to prove measurability on a generating set-system, we get that $f$ is measurable according to the Baire-$sigma$-algebras.







                share|cite|improve this answer












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                share|cite|improve this answer










                answered Nov 22 at 15:55









                p4sch

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