Are Monotone functions Borel Measurable?












14












$begingroup$


Could you guide me how to prove that any monotone function from $Rrightarrow R$ is Borel measurable?
Should we separate the functions into continuous and non-continuous? How to prove for not continuous points?



Thanks for your help






measure-theory







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    I'd try to apply the definition directly. That is, try to show that sets of the form ${xin mathbb{R} | F(x)ge t}$ are Borel.
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:22










  • $begingroup$
    Hi @GiuseppeNegro, I actually have hard time understanding this method. I always use the basic definition of looking at pre-image. Could you explain this a little more. How we show/use this?
    $endgroup$
    – user48405
    Dec 6 '12 at 17:23






  • 1




    $begingroup$
    By definition, a function $mathbb{R}to mathbb{R}$ is Borel-measurable when the preimages of open subsets of $mathbb{R}$ are Borel sets of $mathbb{R}$. Do you agree with this definition?
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:24






  • 1




    $begingroup$
    If you agree, then you can convince yourself that, actually, it is enough to check that the preimages of half-lines are Borel. More precisely, $Fcolon mathbb{R}to mathbb{R}$ is Borel measurable if and only if for every $t in mathbb{R}$ the following set is Borel: $${xin mathbb{R} | F(x)ge t}$$ (Cfr. Rudin, Real and complex analysis, 3rd ed., Theorem 1.12)
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:28










  • $begingroup$
    Yes, it completely matches my definition of $forall B in text{Borel Set} {w: f(w)in B} in F text{ where F is also Borel Set}$
    $endgroup$
    – user48405
    Dec 6 '12 at 17:30


















14












$begingroup$


Could you guide me how to prove that any monotone function from $Rrightarrow R$ is Borel measurable?
Should we separate the functions into continuous and non-continuous? How to prove for not continuous points?



Thanks for your help






measure-theory







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    I'd try to apply the definition directly. That is, try to show that sets of the form ${xin mathbb{R} | F(x)ge t}$ are Borel.
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:22










  • $begingroup$
    Hi @GiuseppeNegro, I actually have hard time understanding this method. I always use the basic definition of looking at pre-image. Could you explain this a little more. How we show/use this?
    $endgroup$
    – user48405
    Dec 6 '12 at 17:23






  • 1




    $begingroup$
    By definition, a function $mathbb{R}to mathbb{R}$ is Borel-measurable when the preimages of open subsets of $mathbb{R}$ are Borel sets of $mathbb{R}$. Do you agree with this definition?
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:24






  • 1




    $begingroup$
    If you agree, then you can convince yourself that, actually, it is enough to check that the preimages of half-lines are Borel. More precisely, $Fcolon mathbb{R}to mathbb{R}$ is Borel measurable if and only if for every $t in mathbb{R}$ the following set is Borel: $${xin mathbb{R} | F(x)ge t}$$ (Cfr. Rudin, Real and complex analysis, 3rd ed., Theorem 1.12)
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:28










  • $begingroup$
    Yes, it completely matches my definition of $forall B in text{Borel Set} {w: f(w)in B} in F text{ where F is also Borel Set}$
    $endgroup$
    – user48405
    Dec 6 '12 at 17:30
















14












14








14


11



$begingroup$


Could you guide me how to prove that any monotone function from $Rrightarrow R$ is Borel measurable?
Should we separate the functions into continuous and non-continuous? How to prove for not continuous points?



Thanks for your help






measure-theory







share|cite|improve this question









$endgroup$




Could you guide me how to prove that any monotone function from $Rrightarrow R$ is Borel measurable?
Should we separate the functions into continuous and non-continuous? How to prove for not continuous points?



Thanks for your help






measure-theory




measure-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 6 '12 at 17:18







user48405















  • 1




    $begingroup$
    I'd try to apply the definition directly. That is, try to show that sets of the form ${xin mathbb{R} | F(x)ge t}$ are Borel.
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:22










  • $begingroup$
    Hi @GiuseppeNegro, I actually have hard time understanding this method. I always use the basic definition of looking at pre-image. Could you explain this a little more. How we show/use this?
    $endgroup$
    – user48405
    Dec 6 '12 at 17:23






  • 1




    $begingroup$
    By definition, a function $mathbb{R}to mathbb{R}$ is Borel-measurable when the preimages of open subsets of $mathbb{R}$ are Borel sets of $mathbb{R}$. Do you agree with this definition?
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:24






  • 1




    $begingroup$
    If you agree, then you can convince yourself that, actually, it is enough to check that the preimages of half-lines are Borel. More precisely, $Fcolon mathbb{R}to mathbb{R}$ is Borel measurable if and only if for every $t in mathbb{R}$ the following set is Borel: $${xin mathbb{R} | F(x)ge t}$$ (Cfr. Rudin, Real and complex analysis, 3rd ed., Theorem 1.12)
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:28










  • $begingroup$
    Yes, it completely matches my definition of $forall B in text{Borel Set} {w: f(w)in B} in F text{ where F is also Borel Set}$
    $endgroup$
    – user48405
    Dec 6 '12 at 17:30
















  • 1




    $begingroup$
    I'd try to apply the definition directly. That is, try to show that sets of the form ${xin mathbb{R} | F(x)ge t}$ are Borel.
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:22










  • $begingroup$
    Hi @GiuseppeNegro, I actually have hard time understanding this method. I always use the basic definition of looking at pre-image. Could you explain this a little more. How we show/use this?
    $endgroup$
    – user48405
    Dec 6 '12 at 17:23






  • 1




    $begingroup$
    By definition, a function $mathbb{R}to mathbb{R}$ is Borel-measurable when the preimages of open subsets of $mathbb{R}$ are Borel sets of $mathbb{R}$. Do you agree with this definition?
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:24






  • 1




    $begingroup$
    If you agree, then you can convince yourself that, actually, it is enough to check that the preimages of half-lines are Borel. More precisely, $Fcolon mathbb{R}to mathbb{R}$ is Borel measurable if and only if for every $t in mathbb{R}$ the following set is Borel: $${xin mathbb{R} | F(x)ge t}$$ (Cfr. Rudin, Real and complex analysis, 3rd ed., Theorem 1.12)
    $endgroup$
    – Giuseppe Negro
    Dec 6 '12 at 17:28










  • $begingroup$
    Yes, it completely matches my definition of $forall B in text{Borel Set} {w: f(w)in B} in F text{ where F is also Borel Set}$
    $endgroup$
    – user48405
    Dec 6 '12 at 17:30










1




1




$begingroup$
I'd try to apply the definition directly. That is, try to show that sets of the form ${xin mathbb{R} | F(x)ge t}$ are Borel.
$endgroup$
– Giuseppe Negro
Dec 6 '12 at 17:22




$begingroup$
I'd try to apply the definition directly. That is, try to show that sets of the form ${xin mathbb{R} | F(x)ge t}$ are Borel.
$endgroup$
– Giuseppe Negro
Dec 6 '12 at 17:22












$begingroup$
Hi @GiuseppeNegro, I actually have hard time understanding this method. I always use the basic definition of looking at pre-image. Could you explain this a little more. How we show/use this?
$endgroup$
– user48405
Dec 6 '12 at 17:23




$begingroup$
Hi @GiuseppeNegro, I actually have hard time understanding this method. I always use the basic definition of looking at pre-image. Could you explain this a little more. How we show/use this?
$endgroup$
– user48405
Dec 6 '12 at 17:23




1




1




$begingroup$
By definition, a function $mathbb{R}to mathbb{R}$ is Borel-measurable when the preimages of open subsets of $mathbb{R}$ are Borel sets of $mathbb{R}$. Do you agree with this definition?
$endgroup$
– Giuseppe Negro
Dec 6 '12 at 17:24




$begingroup$
By definition, a function $mathbb{R}to mathbb{R}$ is Borel-measurable when the preimages of open subsets of $mathbb{R}$ are Borel sets of $mathbb{R}$. Do you agree with this definition?
$endgroup$
– Giuseppe Negro
Dec 6 '12 at 17:24




1




1




$begingroup$
If you agree, then you can convince yourself that, actually, it is enough to check that the preimages of half-lines are Borel. More precisely, $Fcolon mathbb{R}to mathbb{R}$ is Borel measurable if and only if for every $t in mathbb{R}$ the following set is Borel: $${xin mathbb{R} | F(x)ge t}$$ (Cfr. Rudin, Real and complex analysis, 3rd ed., Theorem 1.12)
$endgroup$
– Giuseppe Negro
Dec 6 '12 at 17:28




$begingroup$
If you agree, then you can convince yourself that, actually, it is enough to check that the preimages of half-lines are Borel. More precisely, $Fcolon mathbb{R}to mathbb{R}$ is Borel measurable if and only if for every $t in mathbb{R}$ the following set is Borel: $${xin mathbb{R} | F(x)ge t}$$ (Cfr. Rudin, Real and complex analysis, 3rd ed., Theorem 1.12)
$endgroup$
– Giuseppe Negro
Dec 6 '12 at 17:28












$begingroup$
Yes, it completely matches my definition of $forall B in text{Borel Set} {w: f(w)in B} in F text{ where F is also Borel Set}$
$endgroup$
– user48405
Dec 6 '12 at 17:30






$begingroup$
Yes, it completely matches my definition of $forall B in text{Borel Set} {w: f(w)in B} in F text{ where F is also Borel Set}$
$endgroup$
– user48405
Dec 6 '12 at 17:30












1 Answer
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26












$begingroup$

Hint: If $f$ is monotone, then, for every real number $x$, the set $$f^{-1}((-infty,x])={tmid f(t)leqslant x}$$ is either $varnothing$ or $(-infty,+infty)$ or $(-infty,z)$ or $(-infty,z]$ or $(z,+infty)$ or $[z,+infty)$ for some real number $z$.



To show this, assume for example that $f$ is nondecreasing and that $u$ is in $f^{-1}((-infty,x])$, then show that, for every $vleqslant u$, $v$ is also in $f^{-1}((-infty,x])$.






share|cite|improve this answer











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    26












    $begingroup$

    Hint: If $f$ is monotone, then, for every real number $x$, the set $$f^{-1}((-infty,x])={tmid f(t)leqslant x}$$ is either $varnothing$ or $(-infty,+infty)$ or $(-infty,z)$ or $(-infty,z]$ or $(z,+infty)$ or $[z,+infty)$ for some real number $z$.



    To show this, assume for example that $f$ is nondecreasing and that $u$ is in $f^{-1}((-infty,x])$, then show that, for every $vleqslant u$, $v$ is also in $f^{-1}((-infty,x])$.






    share|cite|improve this answer











    $endgroup$


















      26












      $begingroup$

      Hint: If $f$ is monotone, then, for every real number $x$, the set $$f^{-1}((-infty,x])={tmid f(t)leqslant x}$$ is either $varnothing$ or $(-infty,+infty)$ or $(-infty,z)$ or $(-infty,z]$ or $(z,+infty)$ or $[z,+infty)$ for some real number $z$.



      To show this, assume for example that $f$ is nondecreasing and that $u$ is in $f^{-1}((-infty,x])$, then show that, for every $vleqslant u$, $v$ is also in $f^{-1}((-infty,x])$.






      share|cite|improve this answer











      $endgroup$
















        26












        26








        26





        $begingroup$

        Hint: If $f$ is monotone, then, for every real number $x$, the set $$f^{-1}((-infty,x])={tmid f(t)leqslant x}$$ is either $varnothing$ or $(-infty,+infty)$ or $(-infty,z)$ or $(-infty,z]$ or $(z,+infty)$ or $[z,+infty)$ for some real number $z$.



        To show this, assume for example that $f$ is nondecreasing and that $u$ is in $f^{-1}((-infty,x])$, then show that, for every $vleqslant u$, $v$ is also in $f^{-1}((-infty,x])$.






        share|cite|improve this answer











        $endgroup$



        Hint: If $f$ is monotone, then, for every real number $x$, the set $$f^{-1}((-infty,x])={tmid f(t)leqslant x}$$ is either $varnothing$ or $(-infty,+infty)$ or $(-infty,z)$ or $(-infty,z]$ or $(z,+infty)$ or $[z,+infty)$ for some real number $z$.



        To show this, assume for example that $f$ is nondecreasing and that $u$ is in $f^{-1}((-infty,x])$, then show that, for every $vleqslant u$, $v$ is also in $f^{-1}((-infty,x])$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 11 '18 at 8:15

























        answered Sep 20 '13 at 18:01









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