Prove that $lim_{ntoinfty}int f_n=int f$.











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Let $(f_n)$ be a sequence of non-negative Lebesgue measurable functions such that $f_n$ converges to $f$ and $f_nleq f forall nin mathbb{Z^+}$. It is needed to prove that $lim_{ntoinfty}int f_n=int f$. The following is my solution.



By Fatou's lemma $int f=int (liminf f_n)leq liminfint f_n$. Since $f_nleq f forall nin mathbb{Z^+}$, $int f_nleqint f$. Therefore $limsupint f_nleq int f$. Therefore $limsupint f_nleq int fleq liminfint f_n$. Hence $lim_{ntoinfty}int f_n=int f$.



Could someone tell me if my proof is correct? Thanks.










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  • 2




    Looks good! ${} $
    – PhoemueX
    May 3 '17 at 20:36















up vote
6
down vote

favorite












Let $(f_n)$ be a sequence of non-negative Lebesgue measurable functions such that $f_n$ converges to $f$ and $f_nleq f forall nin mathbb{Z^+}$. It is needed to prove that $lim_{ntoinfty}int f_n=int f$. The following is my solution.



By Fatou's lemma $int f=int (liminf f_n)leq liminfint f_n$. Since $f_nleq f forall nin mathbb{Z^+}$, $int f_nleqint f$. Therefore $limsupint f_nleq int f$. Therefore $limsupint f_nleq int fleq liminfint f_n$. Hence $lim_{ntoinfty}int f_n=int f$.



Could someone tell me if my proof is correct? Thanks.










share|cite|improve this question


















  • 2




    Looks good! ${} $
    – PhoemueX
    May 3 '17 at 20:36













up vote
6
down vote

favorite









up vote
6
down vote

favorite











Let $(f_n)$ be a sequence of non-negative Lebesgue measurable functions such that $f_n$ converges to $f$ and $f_nleq f forall nin mathbb{Z^+}$. It is needed to prove that $lim_{ntoinfty}int f_n=int f$. The following is my solution.



By Fatou's lemma $int f=int (liminf f_n)leq liminfint f_n$. Since $f_nleq f forall nin mathbb{Z^+}$, $int f_nleqint f$. Therefore $limsupint f_nleq int f$. Therefore $limsupint f_nleq int fleq liminfint f_n$. Hence $lim_{ntoinfty}int f_n=int f$.



Could someone tell me if my proof is correct? Thanks.










share|cite|improve this question













Let $(f_n)$ be a sequence of non-negative Lebesgue measurable functions such that $f_n$ converges to $f$ and $f_nleq f forall nin mathbb{Z^+}$. It is needed to prove that $lim_{ntoinfty}int f_n=int f$. The following is my solution.



By Fatou's lemma $int f=int (liminf f_n)leq liminfint f_n$. Since $f_nleq f forall nin mathbb{Z^+}$, $int f_nleqint f$. Therefore $limsupint f_nleq int f$. Therefore $limsupint f_nleq int fleq liminfint f_n$. Hence $lim_{ntoinfty}int f_n=int f$.



Could someone tell me if my proof is correct? Thanks.







measure-theory proof-verification lebesgue-integral self-learning






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asked May 3 '17 at 20:26









Janitha357

1,719516




1,719516








  • 2




    Looks good! ${} $
    – PhoemueX
    May 3 '17 at 20:36














  • 2




    Looks good! ${} $
    – PhoemueX
    May 3 '17 at 20:36








2




2




Looks good! ${} $
– PhoemueX
May 3 '17 at 20:36




Looks good! ${} $
– PhoemueX
May 3 '17 at 20:36










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Your proof is correct! Depending on the level of detail you want to include you might want to consider adding a sentence on why we can deduce that $lim_{n to infty} int f_n$ exists from the fact that $limsup int f_n leq liminf int f_n$.



For future reference, the result you have proved is called Lebesgue's Monotone Convergence Theorem. It is an interesting fact that one can prove the other implication as well, that is, one can derive Fatou's Lemma assuming Lebesgue's Monotone Convergence Theorem. So, if you can write down a proof of LMCT without invoking Fatou's Lemma, then you will have shown that these two results are equivalent to each other. You can take a look at Rudin's Real and Complex Analysis for such a proof of LMCT.






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    up vote
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    Your proof is correct! Depending on the level of detail you want to include you might want to consider adding a sentence on why we can deduce that $lim_{n to infty} int f_n$ exists from the fact that $limsup int f_n leq liminf int f_n$.



    For future reference, the result you have proved is called Lebesgue's Monotone Convergence Theorem. It is an interesting fact that one can prove the other implication as well, that is, one can derive Fatou's Lemma assuming Lebesgue's Monotone Convergence Theorem. So, if you can write down a proof of LMCT without invoking Fatou's Lemma, then you will have shown that these two results are equivalent to each other. You can take a look at Rudin's Real and Complex Analysis for such a proof of LMCT.






    share|cite|improve this answer

























      up vote
      0
      down vote













      Your proof is correct! Depending on the level of detail you want to include you might want to consider adding a sentence on why we can deduce that $lim_{n to infty} int f_n$ exists from the fact that $limsup int f_n leq liminf int f_n$.



      For future reference, the result you have proved is called Lebesgue's Monotone Convergence Theorem. It is an interesting fact that one can prove the other implication as well, that is, one can derive Fatou's Lemma assuming Lebesgue's Monotone Convergence Theorem. So, if you can write down a proof of LMCT without invoking Fatou's Lemma, then you will have shown that these two results are equivalent to each other. You can take a look at Rudin's Real and Complex Analysis for such a proof of LMCT.






      share|cite|improve this answer























        up vote
        0
        down vote










        up vote
        0
        down vote









        Your proof is correct! Depending on the level of detail you want to include you might want to consider adding a sentence on why we can deduce that $lim_{n to infty} int f_n$ exists from the fact that $limsup int f_n leq liminf int f_n$.



        For future reference, the result you have proved is called Lebesgue's Monotone Convergence Theorem. It is an interesting fact that one can prove the other implication as well, that is, one can derive Fatou's Lemma assuming Lebesgue's Monotone Convergence Theorem. So, if you can write down a proof of LMCT without invoking Fatou's Lemma, then you will have shown that these two results are equivalent to each other. You can take a look at Rudin's Real and Complex Analysis for such a proof of LMCT.






        share|cite|improve this answer












        Your proof is correct! Depending on the level of detail you want to include you might want to consider adding a sentence on why we can deduce that $lim_{n to infty} int f_n$ exists from the fact that $limsup int f_n leq liminf int f_n$.



        For future reference, the result you have proved is called Lebesgue's Monotone Convergence Theorem. It is an interesting fact that one can prove the other implication as well, that is, one can derive Fatou's Lemma assuming Lebesgue's Monotone Convergence Theorem. So, if you can write down a proof of LMCT without invoking Fatou's Lemma, then you will have shown that these two results are equivalent to each other. You can take a look at Rudin's Real and Complex Analysis for such a proof of LMCT.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 24 at 6:11









        Brahadeesh

        5,83441958




        5,83441958






























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