function vanishing at infinity and integrability












0












$begingroup$


Suppose $m_n$ is the Lebesgue measure on $mathbb{R}^n$.



Def Let $f:mathbb{R}^nto mathbb{R}$ be continuous. We say that $f$ vanishes at infinity if for every $epsilon>0$ there is a compact $Ksubseteq mathbb{R}^n$ so that $|f|<epsilon$ outside $K$.



Is it true that every continuous $f:mathbb{R}^nto mathbb{R}$ that vanishes at infinity is integrable, ie $$int_{mathbb{R}^n}|f|dm_n<+infty$$ ???



Thanks a lot in advance.






integration measure-theory lebesgue-integral lebesgue-measure







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$endgroup$

















    0












    $begingroup$


    Suppose $m_n$ is the Lebesgue measure on $mathbb{R}^n$.



    Def Let $f:mathbb{R}^nto mathbb{R}$ be continuous. We say that $f$ vanishes at infinity if for every $epsilon>0$ there is a compact $Ksubseteq mathbb{R}^n$ so that $|f|<epsilon$ outside $K$.



    Is it true that every continuous $f:mathbb{R}^nto mathbb{R}$ that vanishes at infinity is integrable, ie $$int_{mathbb{R}^n}|f|dm_n<+infty$$ ???



    Thanks a lot in advance.






    integration measure-theory lebesgue-integral lebesgue-measure







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Suppose $m_n$ is the Lebesgue measure on $mathbb{R}^n$.



      Def Let $f:mathbb{R}^nto mathbb{R}$ be continuous. We say that $f$ vanishes at infinity if for every $epsilon>0$ there is a compact $Ksubseteq mathbb{R}^n$ so that $|f|<epsilon$ outside $K$.



      Is it true that every continuous $f:mathbb{R}^nto mathbb{R}$ that vanishes at infinity is integrable, ie $$int_{mathbb{R}^n}|f|dm_n<+infty$$ ???



      Thanks a lot in advance.






      integration measure-theory lebesgue-integral lebesgue-measure







      share|cite|improve this question









      $endgroup$




      Suppose $m_n$ is the Lebesgue measure on $mathbb{R}^n$.



      Def Let $f:mathbb{R}^nto mathbb{R}$ be continuous. We say that $f$ vanishes at infinity if for every $epsilon>0$ there is a compact $Ksubseteq mathbb{R}^n$ so that $|f|<epsilon$ outside $K$.



      Is it true that every continuous $f:mathbb{R}^nto mathbb{R}$ that vanishes at infinity is integrable, ie $$int_{mathbb{R}^n}|f|dm_n<+infty$$ ???



      Thanks a lot in advance.






      integration measure-theory lebesgue-integral lebesgue-measure




      integration measure-theory lebesgue-integral lebesgue-measure






      share|cite|improve this question













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      asked Dec 12 '18 at 12:35









      eleguitareleguitar

      129114




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          Try $1/(|x|+1)$ in the case $n=1$.






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            Try $1/(|x|+1)$ in the case $n=1$.






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              Try $1/(|x|+1)$ in the case $n=1$.






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                Try $1/(|x|+1)$ in the case $n=1$.






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                Try $1/(|x|+1)$ in the case $n=1$.







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                answered Dec 12 '18 at 12:44









                Robert IsraelRobert Israel

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