Two different measures with equal support
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Can there be two different complex (hence finite) measures whose support is equal? I presume no, as it may defy some separation theorems otherwise. But I need a concrete proof. Any help is hugely appreciated !
functional-analysis measure-theory
asked Dec 10 '18 at 21:03
NewUserNewUser
908
$endgroup$
add a comment |
$begingroup$
Can there be two different complex (hence finite) measures whose support is equal? I presume no, as it may defy some separation theorems otherwise. But I need a concrete proof. Any help is hugely appreciated !
functional-analysis measure-theory
asked Dec 10 '18 at 21:03
NewUserNewUser
908
$endgroup$
$begingroup$
Multiply one measure by some density which is bounded from away and bounded away from zero.
$endgroup$
– Dirk
Dec 10 '18 at 21:19
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What is your definition of support?
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– DisintegratingByParts
Dec 10 '18 at 21:44
add a comment |
$begingroup$
Can there be two different complex (hence finite) measures whose support is equal? I presume no, as it may defy some separation theorems otherwise. But I need a concrete proof. Any help is hugely appreciated !
functional-analysis measure-theory
asked Dec 10 '18 at 21:03
NewUserNewUser
908
$endgroup$
Can there be two different complex (hence finite) measures whose support is equal? I presume no, as it may defy some separation theorems otherwise. But I need a concrete proof. Any help is hugely appreciated !
functional-analysis measure-theory
functional-analysis measure-theory
asked Dec 10 '18 at 21:03
NewUserNewUser
908
asked Dec 10 '18 at 21:03
NewUserNewUser
908
asked Dec 10 '18 at 21:03
NewUserNewUser
908
asked Dec 10 '18 at 21:03
NewUserNewUser
908
asked Dec 10 '18 at 21:03
NewUserNewUser
908
908
$begingroup$
Multiply one measure by some density which is bounded from away and bounded away from zero.
$endgroup$
– Dirk
Dec 10 '18 at 21:19
$begingroup$
What is your definition of support?
$endgroup$
– DisintegratingByParts
Dec 10 '18 at 21:44
add a comment |
$begingroup$
Multiply one measure by some density which is bounded from away and bounded away from zero.
$endgroup$
– Dirk
Dec 10 '18 at 21:19
$begingroup$
What is your definition of support?
$endgroup$
– DisintegratingByParts
Dec 10 '18 at 21:44
$begingroup$
Multiply one measure by some density which is bounded from away and bounded away from zero.
$endgroup$
– Dirk
Dec 10 '18 at 21:19
$begingroup$
Multiply one measure by some density which is bounded from away and bounded away from zero.
$endgroup$
– Dirk
Dec 10 '18 at 21:19
$begingroup$
What is your definition of support?
$endgroup$
– DisintegratingByParts
Dec 10 '18 at 21:44
$begingroup$
What is your definition of support?
$endgroup$
– DisintegratingByParts
Dec 10 '18 at 21:44
add a comment |
1 Answer
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votes
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The support of a measure $mu$ is the complement of the largest open set $U$ with $mu (U)=0$. If $f$ and $g$ are strictly positive integrable functions on $mathbb R$ and $mu (A)=int_A f(x)dx,nu (A)=int_A g(x)dx$ then the support of either measure is $mathbb R$ (because the only open set $U$ of measure $0$ is the empty set). Take $f(x)=e^{-|x|}, g(x)=e^{-2|x|}$, for example to get a counterexample.
answered Dec 10 '18 at 23:45
Kavi Rama MurthyKavi Rama Murthy
58.7k42161
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1 Answer
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1 Answer
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active
oldest
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active
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votes
$begingroup$
The support of a measure $mu$ is the complement of the largest open set $U$ with $mu (U)=0$. If $f$ and $g$ are strictly positive integrable functions on $mathbb R$ and $mu (A)=int_A f(x)dx,nu (A)=int_A g(x)dx$ then the support of either measure is $mathbb R$ (because the only open set $U$ of measure $0$ is the empty set). Take $f(x)=e^{-|x|}, g(x)=e^{-2|x|}$, for example to get a counterexample.
answered Dec 10 '18 at 23:45
Kavi Rama MurthyKavi Rama Murthy
58.7k42161
$endgroup$
add a comment |
$begingroup$
The support of a measure $mu$ is the complement of the largest open set $U$ with $mu (U)=0$. If $f$ and $g$ are strictly positive integrable functions on $mathbb R$ and $mu (A)=int_A f(x)dx,nu (A)=int_A g(x)dx$ then the support of either measure is $mathbb R$ (because the only open set $U$ of measure $0$ is the empty set). Take $f(x)=e^{-|x|}, g(x)=e^{-2|x|}$, for example to get a counterexample.
answered Dec 10 '18 at 23:45
Kavi Rama MurthyKavi Rama Murthy
58.7k42161
$endgroup$
add a comment |
$begingroup$
The support of a measure $mu$ is the complement of the largest open set $U$ with $mu (U)=0$. If $f$ and $g$ are strictly positive integrable functions on $mathbb R$ and $mu (A)=int_A f(x)dx,nu (A)=int_A g(x)dx$ then the support of either measure is $mathbb R$ (because the only open set $U$ of measure $0$ is the empty set). Take $f(x)=e^{-|x|}, g(x)=e^{-2|x|}$, for example to get a counterexample.
answered Dec 10 '18 at 23:45
Kavi Rama MurthyKavi Rama Murthy
58.7k42161
$endgroup$
The support of a measure $mu$ is the complement of the largest open set $U$ with $mu (U)=0$. If $f$ and $g$ are strictly positive integrable functions on $mathbb R$ and $mu (A)=int_A f(x)dx,nu (A)=int_A g(x)dx$ then the support of either measure is $mathbb R$ (because the only open set $U$ of measure $0$ is the empty set). Take $f(x)=e^{-|x|}, g(x)=e^{-2|x|}$, for example to get a counterexample.
answered Dec 10 '18 at 23:45
Kavi Rama MurthyKavi Rama Murthy
58.7k42161
answered Dec 10 '18 at 23:45
Kavi Rama MurthyKavi Rama Murthy
58.7k42161
answered Dec 10 '18 at 23:45
Kavi Rama MurthyKavi Rama Murthy
58.7k42161
answered Dec 10 '18 at 23:45
Kavi Rama MurthyKavi Rama Murthy
58.7k42161
58.7k42161
add a comment |
add a comment |
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$begingroup$
Multiply one measure by some density which is bounded from away and bounded away from zero.
$endgroup$
– Dirk
Dec 10 '18 at 21:19
$begingroup$
What is your definition of support?
$endgroup$
– DisintegratingByParts
Dec 10 '18 at 21:44