Doob Dynkin lemma, uniqueness of the Borel function












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For $X$ and $Y$ defined on some probability space $(Omega,mathcal{F},mathbf{P})$, $Y$ is $sigma(X)$-measurable iff there is a Borel function $f$ such that $Y=f(X)$.



My question is, given $X$ and $Y$ and knowing that $Y$ being $sigma(X)$-measurable, is the Borel function $f$ unique? Can there be more than one such Borel functions fulfill the condition?










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    0














    For $X$ and $Y$ defined on some probability space $(Omega,mathcal{F},mathbf{P})$, $Y$ is $sigma(X)$-measurable iff there is a Borel function $f$ such that $Y=f(X)$.



    My question is, given $X$ and $Y$ and knowing that $Y$ being $sigma(X)$-measurable, is the Borel function $f$ unique? Can there be more than one such Borel functions fulfill the condition?










    share|cite|improve this question



























      0












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      0





      For $X$ and $Y$ defined on some probability space $(Omega,mathcal{F},mathbf{P})$, $Y$ is $sigma(X)$-measurable iff there is a Borel function $f$ such that $Y=f(X)$.



      My question is, given $X$ and $Y$ and knowing that $Y$ being $sigma(X)$-measurable, is the Borel function $f$ unique? Can there be more than one such Borel functions fulfill the condition?










      share|cite|improve this question















      For $X$ and $Y$ defined on some probability space $(Omega,mathcal{F},mathbf{P})$, $Y$ is $sigma(X)$-measurable iff there is a Borel function $f$ such that $Y=f(X)$.



      My question is, given $X$ and $Y$ and knowing that $Y$ being $sigma(X)$-measurable, is the Borel function $f$ unique? Can there be more than one such Borel functions fulfill the condition?







      probability-theory measure-theory






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      share|cite|improve this question













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      edited Nov 19 '18 at 10:23

























      asked Nov 19 '18 at 1:10









      lychtalent

      619




      619






















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          It helps to rephrase this question in terms of pure measure theory: let measurable $X,YinOmegatoPsi$ be such that $Y=fcirc X$ for some Borel $f$. Is $f$ unique?



          This immediately makes it clear that Borelness is not relevant to this problem. The answer is the well-known theory of lifts in the category of sets.



          If $X$ is not surjective, the answer is clearly no; we can make arbitrary changes to $f$ on the complement of $text{im}{(X)}$.



          If $X$ is surjective, then $f$ is unique by linearity. For, suppose $Y=fcirc X=gcirc X$; then, by subtracting, we have $(f-g)circ X=0$. As $X$ is surjective, $f-g=0$.






          share|cite|improve this answer





















          • Thanks! This is an excellent answer!
            – lychtalent
            Nov 19 '18 at 10:23











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          1 Answer
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          active

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2














          It helps to rephrase this question in terms of pure measure theory: let measurable $X,YinOmegatoPsi$ be such that $Y=fcirc X$ for some Borel $f$. Is $f$ unique?



          This immediately makes it clear that Borelness is not relevant to this problem. The answer is the well-known theory of lifts in the category of sets.



          If $X$ is not surjective, the answer is clearly no; we can make arbitrary changes to $f$ on the complement of $text{im}{(X)}$.



          If $X$ is surjective, then $f$ is unique by linearity. For, suppose $Y=fcirc X=gcirc X$; then, by subtracting, we have $(f-g)circ X=0$. As $X$ is surjective, $f-g=0$.






          share|cite|improve this answer





















          • Thanks! This is an excellent answer!
            – lychtalent
            Nov 19 '18 at 10:23
















          2














          It helps to rephrase this question in terms of pure measure theory: let measurable $X,YinOmegatoPsi$ be such that $Y=fcirc X$ for some Borel $f$. Is $f$ unique?



          This immediately makes it clear that Borelness is not relevant to this problem. The answer is the well-known theory of lifts in the category of sets.



          If $X$ is not surjective, the answer is clearly no; we can make arbitrary changes to $f$ on the complement of $text{im}{(X)}$.



          If $X$ is surjective, then $f$ is unique by linearity. For, suppose $Y=fcirc X=gcirc X$; then, by subtracting, we have $(f-g)circ X=0$. As $X$ is surjective, $f-g=0$.






          share|cite|improve this answer





















          • Thanks! This is an excellent answer!
            – lychtalent
            Nov 19 '18 at 10:23














          2












          2








          2






          It helps to rephrase this question in terms of pure measure theory: let measurable $X,YinOmegatoPsi$ be such that $Y=fcirc X$ for some Borel $f$. Is $f$ unique?



          This immediately makes it clear that Borelness is not relevant to this problem. The answer is the well-known theory of lifts in the category of sets.



          If $X$ is not surjective, the answer is clearly no; we can make arbitrary changes to $f$ on the complement of $text{im}{(X)}$.



          If $X$ is surjective, then $f$ is unique by linearity. For, suppose $Y=fcirc X=gcirc X$; then, by subtracting, we have $(f-g)circ X=0$. As $X$ is surjective, $f-g=0$.






          share|cite|improve this answer












          It helps to rephrase this question in terms of pure measure theory: let measurable $X,YinOmegatoPsi$ be such that $Y=fcirc X$ for some Borel $f$. Is $f$ unique?



          This immediately makes it clear that Borelness is not relevant to this problem. The answer is the well-known theory of lifts in the category of sets.



          If $X$ is not surjective, the answer is clearly no; we can make arbitrary changes to $f$ on the complement of $text{im}{(X)}$.



          If $X$ is surjective, then $f$ is unique by linearity. For, suppose $Y=fcirc X=gcirc X$; then, by subtracting, we have $(f-g)circ X=0$. As $X$ is surjective, $f-g=0$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 19 '18 at 1:22









          Jacob Manaker

          1,119414




          1,119414












          • Thanks! This is an excellent answer!
            – lychtalent
            Nov 19 '18 at 10:23


















          • Thanks! This is an excellent answer!
            – lychtalent
            Nov 19 '18 at 10:23
















          Thanks! This is an excellent answer!
          – lychtalent
          Nov 19 '18 at 10:23




          Thanks! This is an excellent answer!
          – lychtalent
          Nov 19 '18 at 10:23


















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