Show that $f$ is measurable on $(a,b)$











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Let $f,F : (a,b)to mathbb{R}$ such that $F$ is differentiable on $(a,b)$ and $F'(x) =f(x)$ for each $x in (a,b)$. Show that $f$ is measurable on $(a,b)$.
Hint : Show first that $forall x in (a,b) : f(x) = limlimits_{n to infty} n[F(x+frac{1}{n}) - F(x)]$.



I have already proved the hint ... How to proceed please?










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    Let $f,F : (a,b)to mathbb{R}$ such that $F$ is differentiable on $(a,b)$ and $F'(x) =f(x)$ for each $x in (a,b)$. Show that $f$ is measurable on $(a,b)$.
    Hint : Show first that $forall x in (a,b) : f(x) = limlimits_{n to infty} n[F(x+frac{1}{n}) - F(x)]$.



    I have already proved the hint ... How to proceed please?










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      down vote

      favorite









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      down vote

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      Let $f,F : (a,b)to mathbb{R}$ such that $F$ is differentiable on $(a,b)$ and $F'(x) =f(x)$ for each $x in (a,b)$. Show that $f$ is measurable on $(a,b)$.
      Hint : Show first that $forall x in (a,b) : f(x) = limlimits_{n to infty} n[F(x+frac{1}{n}) - F(x)]$.



      I have already proved the hint ... How to proceed please?










      share|cite|improve this question













      Let $f,F : (a,b)to mathbb{R}$ such that $F$ is differentiable on $(a,b)$ and $F'(x) =f(x)$ for each $x in (a,b)$. Show that $f$ is measurable on $(a,b)$.
      Hint : Show first that $forall x in (a,b) : f(x) = limlimits_{n to infty} n[F(x+frac{1}{n}) - F(x)]$.



      I have already proved the hint ... How to proceed please?







      measurable-functions






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      asked Nov 23 at 22:01









      Ahmed

      1,244511




      1,244511






















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          Note that each function $xmapsto n(F(x + 1/n) - F(x))$ is measurable and that the limit of a sequence of measurable functions is measurable.






          share|cite|improve this answer





















          • We can say this because : Since F is differentiable then it is continuous and so it is measurable. Isn't it?
            – Ahmed
            Nov 23 at 22:14








          • 1




            That is correct.
            – ncmathsadist
            Nov 23 at 22:17











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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

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          up vote
          0
          down vote



          accepted










          Note that each function $xmapsto n(F(x + 1/n) - F(x))$ is measurable and that the limit of a sequence of measurable functions is measurable.






          share|cite|improve this answer





















          • We can say this because : Since F is differentiable then it is continuous and so it is measurable. Isn't it?
            – Ahmed
            Nov 23 at 22:14








          • 1




            That is correct.
            – ncmathsadist
            Nov 23 at 22:17















          up vote
          0
          down vote



          accepted










          Note that each function $xmapsto n(F(x + 1/n) - F(x))$ is measurable and that the limit of a sequence of measurable functions is measurable.






          share|cite|improve this answer





















          • We can say this because : Since F is differentiable then it is continuous and so it is measurable. Isn't it?
            – Ahmed
            Nov 23 at 22:14








          • 1




            That is correct.
            – ncmathsadist
            Nov 23 at 22:17













          up vote
          0
          down vote



          accepted







          up vote
          0
          down vote



          accepted






          Note that each function $xmapsto n(F(x + 1/n) - F(x))$ is measurable and that the limit of a sequence of measurable functions is measurable.






          share|cite|improve this answer












          Note that each function $xmapsto n(F(x + 1/n) - F(x))$ is measurable and that the limit of a sequence of measurable functions is measurable.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 23 at 22:04









          ncmathsadist

          41.9k259101




          41.9k259101












          • We can say this because : Since F is differentiable then it is continuous and so it is measurable. Isn't it?
            – Ahmed
            Nov 23 at 22:14








          • 1




            That is correct.
            – ncmathsadist
            Nov 23 at 22:17


















          • We can say this because : Since F is differentiable then it is continuous and so it is measurable. Isn't it?
            – Ahmed
            Nov 23 at 22:14








          • 1




            That is correct.
            – ncmathsadist
            Nov 23 at 22:17
















          We can say this because : Since F is differentiable then it is continuous and so it is measurable. Isn't it?
          – Ahmed
          Nov 23 at 22:14






          We can say this because : Since F is differentiable then it is continuous and so it is measurable. Isn't it?
          – Ahmed
          Nov 23 at 22:14






          1




          1




          That is correct.
          – ncmathsadist
          Nov 23 at 22:17




          That is correct.
          – ncmathsadist
          Nov 23 at 22:17


















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