function vanishing at infinity and integrability
$begingroup$
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
asked Dec 12 '18 at 12:35
eleguitareleguitar
129114
$endgroup$
add a comment |
$begingroup$
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
asked Dec 12 '18 at 12:35
eleguitareleguitar
129114
$endgroup$
add a comment |
$begingroup$
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
asked Dec 12 '18 at 12:35
eleguitareleguitar
129114
$endgroup$
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
asked Dec 12 '18 at 12:35
eleguitareleguitar
129114
asked Dec 12 '18 at 12:35
eleguitareleguitar
129114
asked Dec 12 '18 at 12:35
eleguitareleguitar
129114
asked Dec 12 '18 at 12:35
eleguitareleguitar
129114
asked Dec 12 '18 at 12:35
eleguitareleguitar
129114
129114
add a comment |
add a comment |
1 Answer
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$begingroup$
Try $1/(|x|+1)$ in the case $n=1$.
answered Dec 12 '18 at 12:44
Robert IsraelRobert Israel
323k23212466
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1 Answer
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1 Answer
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$begingroup$
Try $1/(|x|+1)$ in the case $n=1$.
answered Dec 12 '18 at 12:44
Robert IsraelRobert Israel
323k23212466
$endgroup$
add a comment |
$begingroup$
Try $1/(|x|+1)$ in the case $n=1$.
answered Dec 12 '18 at 12:44
Robert IsraelRobert Israel
323k23212466
$endgroup$
add a comment |
$begingroup$
Try $1/(|x|+1)$ in the case $n=1$.
answered Dec 12 '18 at 12:44
Robert IsraelRobert Israel
323k23212466
$endgroup$
Try $1/(|x|+1)$ in the case $n=1$.
answered Dec 12 '18 at 12:44
Robert IsraelRobert Israel
323k23212466
answered Dec 12 '18 at 12:44
Robert IsraelRobert Israel
323k23212466
answered Dec 12 '18 at 12:44
Robert IsraelRobert Israel
323k23212466
answered Dec 12 '18 at 12:44
Robert IsraelRobert Israel
323k23212466
323k23212466
add a comment |
add a comment |
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