Sequence such that $E_n to E$ and $m(cup E_n) = infty$ then $lim_n m(E_n) > m(lim_n E_n)$ (Example)

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I proved that




If $(E_n)$ is a sequence of measurable sets such that $E_n to E$ and $m(cup E_n) < infty$, then $lim_n m(E_n) = m(lim_n E_n)$.




Now, I'm trying to find a example when $E_n to E$ and $m(cup E_n) = infty$ such that $lim_n m(E_n) > m(lim_n E_n)$ but I didn't succeed. Can anybody help me?






real-analysis measure-theory lebesgue-measure







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  • What is meant by $E_nto E$?
    – amsmath
    Sep 1 at 0:37










  • @amsmath, sorry for the terrible notation. I want meant "$(E_n)$ converges"
    – Lucas Corrêa
    Sep 1 at 0:39










  • What is meant by this? Do you mean $E_1subset E_2subsetldots$ and $E = bigcup_n E_n$?
    – amsmath
    Sep 1 at 0:39







  • 1




    @amsmath This is fairly standard. $E_n to E$ means $lim sup E_n =lim inf E_n =E$.
    – Kavi Rama Murthy
    Sep 1 at 0:41







  • 1




    @amsmath Or (exercise) equivalently, that $chi_E_ntochi_E$ pointwise.
    – David C. Ullrich
    Sep 1 at 1:15














up vote
0
down vote

favorite












I proved that




If $(E_n)$ is a sequence of measurable sets such that $E_n to E$ and $m(cup E_n) < infty$, then $lim_n m(E_n) = m(lim_n E_n)$.




Now, I'm trying to find a example when $E_n to E$ and $m(cup E_n) = infty$ such that $lim_n m(E_n) > m(lim_n E_n)$ but I didn't succeed. Can anybody help me?






real-analysis measure-theory lebesgue-measure







share|cite|improve this question





















  • What is meant by $E_nto E$?
    – amsmath
    Sep 1 at 0:37










  • @amsmath, sorry for the terrible notation. I want meant "$(E_n)$ converges"
    – Lucas Corrêa
    Sep 1 at 0:39










  • What is meant by this? Do you mean $E_1subset E_2subsetldots$ and $E = bigcup_n E_n$?
    – amsmath
    Sep 1 at 0:39







  • 1




    @amsmath This is fairly standard. $E_n to E$ means $lim sup E_n =lim inf E_n =E$.
    – Kavi Rama Murthy
    Sep 1 at 0:41







  • 1




    @amsmath Or (exercise) equivalently, that $chi_E_ntochi_E$ pointwise.
    – David C. Ullrich
    Sep 1 at 1:15












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I proved that




If $(E_n)$ is a sequence of measurable sets such that $E_n to E$ and $m(cup E_n) < infty$, then $lim_n m(E_n) = m(lim_n E_n)$.




Now, I'm trying to find a example when $E_n to E$ and $m(cup E_n) = infty$ such that $lim_n m(E_n) > m(lim_n E_n)$ but I didn't succeed. Can anybody help me?






real-analysis measure-theory lebesgue-measure







share|cite|improve this question













I proved that




If $(E_n)$ is a sequence of measurable sets such that $E_n to E$ and $m(cup E_n) < infty$, then $lim_n m(E_n) = m(lim_n E_n)$.




Now, I'm trying to find a example when $E_n to E$ and $m(cup E_n) = infty$ such that $lim_n m(E_n) > m(lim_n E_n)$ but I didn't succeed. Can anybody help me?






real-analysis measure-theory lebesgue-measure




real-analysis measure-theory lebesgue-measure






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Sep 1 at 0:24









Lucas Corrêa

1,128319




1,128319











  • What is meant by $E_nto E$?
    – amsmath
    Sep 1 at 0:37










  • @amsmath, sorry for the terrible notation. I want meant "$(E_n)$ converges"
    – Lucas Corrêa
    Sep 1 at 0:39










  • What is meant by this? Do you mean $E_1subset E_2subsetldots$ and $E = bigcup_n E_n$?
    – amsmath
    Sep 1 at 0:39







  • 1




    @amsmath This is fairly standard. $E_n to E$ means $lim sup E_n =lim inf E_n =E$.
    – Kavi Rama Murthy
    Sep 1 at 0:41







  • 1




    @amsmath Or (exercise) equivalently, that $chi_E_ntochi_E$ pointwise.
    – David C. Ullrich
    Sep 1 at 1:15
















  • What is meant by $E_nto E$?
    – amsmath
    Sep 1 at 0:37










  • @amsmath, sorry for the terrible notation. I want meant "$(E_n)$ converges"
    – Lucas Corrêa
    Sep 1 at 0:39










  • What is meant by this? Do you mean $E_1subset E_2subsetldots$ and $E = bigcup_n E_n$?
    – amsmath
    Sep 1 at 0:39







  • 1




    @amsmath This is fairly standard. $E_n to E$ means $lim sup E_n =lim inf E_n =E$.
    – Kavi Rama Murthy
    Sep 1 at 0:41







  • 1




    @amsmath Or (exercise) equivalently, that $chi_E_ntochi_E$ pointwise.
    – David C. Ullrich
    Sep 1 at 1:15















What is meant by $E_nto E$?
– amsmath
Sep 1 at 0:37




What is meant by $E_nto E$?
– amsmath
Sep 1 at 0:37












@amsmath, sorry for the terrible notation. I want meant "$(E_n)$ converges"
– Lucas Corrêa
Sep 1 at 0:39




@amsmath, sorry for the terrible notation. I want meant "$(E_n)$ converges"
– Lucas Corrêa
Sep 1 at 0:39












What is meant by this? Do you mean $E_1subset E_2subsetldots$ and $E = bigcup_n E_n$?
– amsmath
Sep 1 at 0:39





What is meant by this? Do you mean $E_1subset E_2subsetldots$ and $E = bigcup_n E_n$?
– amsmath
Sep 1 at 0:39





1




1




@amsmath This is fairly standard. $E_n to E$ means $lim sup E_n =lim inf E_n =E$.
– Kavi Rama Murthy
Sep 1 at 0:41





@amsmath This is fairly standard. $E_n to E$ means $lim sup E_n =lim inf E_n =E$.
– Kavi Rama Murthy
Sep 1 at 0:41





1




1




@amsmath Or (exercise) equivalently, that $chi_E_ntochi_E$ pointwise.
– David C. Ullrich
Sep 1 at 1:15




@amsmath Or (exercise) equivalently, that $chi_E_ntochi_E$ pointwise.
– David C. Ullrich
Sep 1 at 1:15










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$E_n=[n,infty)$ is an obvious example.






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    1 Answer
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    active

    oldest

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    active

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    active

    oldest

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    up vote
    3
    down vote



    accepted










    $E_n=[n,infty)$ is an obvious example.






    share|cite|improve this answer
























      up vote
      3
      down vote



      accepted










      $E_n=[n,infty)$ is an obvious example.






      share|cite|improve this answer






















        up vote
        3
        down vote



        accepted







        up vote
        3
        down vote



        accepted






        $E_n=[n,infty)$ is an obvious example.






        share|cite|improve this answer












        $E_n=[n,infty)$ is an obvious example.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Sep 1 at 0:39









        Kavi Rama Murthy

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