If $int f=0$ then $f=0$ a.e. with $fgeq 0$. Is it true if $f(x) = infty$?
0
$begingroup$
I have a doubt!
I know that if $f$ measurable and nonnegative, $int f=0$ implies $f=0$ a.e.
And if $m(E)=0$ then $int_{E}f=0$ (even if $f(x)=infty$ forall $x$)
If $f(x)=infty$ forall $x$, $f:Xto overline{mathbb{R}}$ and $m(X)=0$ then $int_{X}f=0$ implies $f=0$ a.e.
Can it be $f = 0$ a.e. even when $f(x) = infty$ for all $x$ in $X$?
real-analysis measure-theory lebesgue-integral lebesgue-measure
asked Dec 7 '18 at 23:26
eraldcoileraldcoil
385211
$endgroup$
add a comment |
0
$begingroup$
I have a doubt!
I know that if $f$ measurable and nonnegative, $int f=0$ implies $f=0$ a.e.
And if $m(E)=0$ then $int_{E}f=0$ (even if $f(x)=infty$ forall $x$)
If $f(x)=infty$ forall $x$, $f:Xto overline{mathbb{R}}$ and $m(X)=0$ then $int_{X}f=0$ implies $f=0$ a.e.
Can it be $f = 0$ a.e. even when $f(x) = infty$ for all $x$ in $X$?
real-analysis measure-theory lebesgue-integral lebesgue-measure
asked Dec 7 '18 at 23:26
eraldcoileraldcoil
385211
$endgroup$
1
$begingroup$
Yes, since $X$ has measure zero. It may sound strange, however if the space has measure zero itself then this space endowed with that measure does not have a measure theoretic interest.
$endgroup$
– clark
Dec 7 '18 at 23:31
$begingroup$
In particular, if $m(X) = 0$, then any (syntactically valid) statement in terms of $x$ is true for a.e. $x in X$.
$endgroup$
– Daniel Schepler
Dec 7 '18 at 23:37
$begingroup$
When stating a result, you must put the conditions before the conclusions! This is a common error. Your title should read: "If $int f=0$ with $fge 0$, then $f=0$ a.e."
$endgroup$
– TonyK
Dec 7 '18 at 23:39
$begingroup$
What do you mean by $f(x)=infty$? It is possible to give it a meaning, but not, I think, in the context of Lebesgue integration.
$endgroup$
– TonyK
Dec 8 '18 at 21:04
add a comment |
0
0
0
$begingroup$
I have a doubt!
I know that if $f$ measurable and nonnegative, $int f=0$ implies $f=0$ a.e.
And if $m(E)=0$ then $int_{E}f=0$ (even if $f(x)=infty$ forall $x$)
If $f(x)=infty$ forall $x$, $f:Xto overline{mathbb{R}}$ and $m(X)=0$ then $int_{X}f=0$ implies $f=0$ a.e.
Can it be $f = 0$ a.e. even when $f(x) = infty$ for all $x$ in $X$?
real-analysis measure-theory lebesgue-integral lebesgue-measure
asked Dec 7 '18 at 23:26
eraldcoileraldcoil
385211
$endgroup$
I have a doubt!
I know that if $f$ measurable and nonnegative, $int f=0$ implies $f=0$ a.e.
And if $m(E)=0$ then $int_{E}f=0$ (even if $f(x)=infty$ forall $x$)
If $f(x)=infty$ forall $x$, $f:Xto overline{mathbb{R}}$ and $m(X)=0$ then $int_{X}f=0$ implies $f=0$ a.e.
Can it be $f = 0$ a.e. even when $f(x) = infty$ for all $x$ in $X$?
real-analysis measure-theory lebesgue-integral lebesgue-measure
real-analysis measure-theory lebesgue-integral lebesgue-measure
asked Dec 7 '18 at 23:26
eraldcoileraldcoil
385211
asked Dec 7 '18 at 23:26
eraldcoileraldcoil
385211
asked Dec 7 '18 at 23:26
eraldcoileraldcoil
385211
asked Dec 7 '18 at 23:26
eraldcoileraldcoil
385211
asked Dec 7 '18 at 23:26
eraldcoileraldcoil
385211
385211
1
$begingroup$
Yes, since $X$ has measure zero. It may sound strange, however if the space has measure zero itself then this space endowed with that measure does not have a measure theoretic interest.
$endgroup$
– clark
Dec 7 '18 at 23:31
$begingroup$
In particular, if $m(X) = 0$, then any (syntactically valid) statement in terms of $x$ is true for a.e. $x in X$.
$endgroup$
– Daniel Schepler
Dec 7 '18 at 23:37
$begingroup$
When stating a result, you must put the conditions before the conclusions! This is a common error. Your title should read: "If $int f=0$ with $fge 0$, then $f=0$ a.e."
$endgroup$
– TonyK
Dec 7 '18 at 23:39
$begingroup$
What do you mean by $f(x)=infty$? It is possible to give it a meaning, but not, I think, in the context of Lebesgue integration.
$endgroup$
– TonyK
Dec 8 '18 at 21:04
add a comment |
1
$begingroup$
Yes, since $X$ has measure zero. It may sound strange, however if the space has measure zero itself then this space endowed with that measure does not have a measure theoretic interest.
$endgroup$
– clark
Dec 7 '18 at 23:31
$begingroup$
In particular, if $m(X) = 0$, then any (syntactically valid) statement in terms of $x$ is true for a.e. $x in X$.
$endgroup$
– Daniel Schepler
Dec 7 '18 at 23:37
$begingroup$
When stating a result, you must put the conditions before the conclusions! This is a common error. Your title should read: "If $int f=0$ with $fge 0$, then $f=0$ a.e."
$endgroup$
– TonyK
Dec 7 '18 at 23:39
$begingroup$
What do you mean by $f(x)=infty$? It is possible to give it a meaning, but not, I think, in the context of Lebesgue integration.
$endgroup$
– TonyK
Dec 8 '18 at 21:04
1
1
$begingroup$
Yes, since $X$ has measure zero. It may sound strange, however if the space has measure zero itself then this space endowed with that measure does not have a measure theoretic interest.
$endgroup$
– clark
Dec 7 '18 at 23:31
$begingroup$
Yes, since $X$ has measure zero. It may sound strange, however if the space has measure zero itself then this space endowed with that measure does not have a measure theoretic interest.
$endgroup$
– clark
Dec 7 '18 at 23:31
$begingroup$
In particular, if $m(X) = 0$, then any (syntactically valid) statement in terms of $x$ is true for a.e. $x in X$.
$endgroup$
– Daniel Schepler
Dec 7 '18 at 23:37
$begingroup$
In particular, if $m(X) = 0$, then any (syntactically valid) statement in terms of $x$ is true for a.e. $x in X$.
$endgroup$
– Daniel Schepler
Dec 7 '18 at 23:37
$begingroup$
When stating a result, you must put the conditions before the conclusions! This is a common error. Your title should read: "If $int f=0$ with $fge 0$, then $f=0$ a.e."
$endgroup$
– TonyK
Dec 7 '18 at 23:39
$begingroup$
When stating a result, you must put the conditions before the conclusions! This is a common error. Your title should read: "If $int f=0$ with $fge 0$, then $f=0$ a.e."
$endgroup$
– TonyK
Dec 7 '18 at 23:39
$begingroup$
What do you mean by $f(x)=infty$? It is possible to give it a meaning, but not, I think, in the context of Lebesgue integration.
$endgroup$
– TonyK
Dec 8 '18 at 21:04
$begingroup$
What do you mean by $f(x)=infty$? It is possible to give it a meaning, but not, I think, in the context of Lebesgue integration.
$endgroup$
– TonyK
Dec 8 '18 at 21:04
add a comment |
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1
$begingroup$
Yes, since $X$ has measure zero. It may sound strange, however if the space has measure zero itself then this space endowed with that measure does not have a measure theoretic interest.
$endgroup$
– clark
Dec 7 '18 at 23:31
$begingroup$
In particular, if $m(X) = 0$, then any (syntactically valid) statement in terms of $x$ is true for a.e. $x in X$.
$endgroup$
– Daniel Schepler
Dec 7 '18 at 23:37
$begingroup$
When stating a result, you must put the conditions before the conclusions! This is a common error. Your title should read: "If $int f=0$ with $fge 0$, then $f=0$ a.e."
$endgroup$
– TonyK
Dec 7 '18 at 23:39
$begingroup$
What do you mean by $f(x)=infty$? It is possible to give it a meaning, but not, I think, in the context of Lebesgue integration.
$endgroup$
– TonyK
Dec 8 '18 at 21:04