Supremum over integral of bounded and continuous functions is $+infty$Fubini's Theorem and bounded...
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Supremum over integral of bounded and continuous functions is $+infty$
Fubini's Theorem and bounded functionsContinuity of a probability integral with supremumBounded Lipschitz Metric on Space of Positive MeasuresCountable additivity of measure with given “good” marginalsShow that $iint_{X times Y}varphi(x)k(x,y)psi(y) d(mu times nu)=int_Y Big[int_Xvarphi(x)k(x,y)dmu Big] psi(y) dnu$weak convergence and unbounded functions with bounded momentLink between convergence in Probability of a supremum and a limsupIs the set of all joint probability measure closed?Realize a coupling in the target space via a measure on the source spaceIs an ambiguity set with Wasserstein distance of order 1 is convex?
$begingroup$
Let $X, Y$ be Polish spaces with probability measures $mu, nu$, respectively; and let $gamma$ be a finite measure on the product $X times Y$. We consider the maximization problem
$$
sup_{varphi,psi} bigg{int_{X} varphi dmu + int_{Y} psi dnu - int_{X times Y} big( varphi(x) + psi(y) big) dgammabigg}
$$
over the set of all bounded and continuous functions $varphi(x),psi(y)$.
Why can this maximization problem always attain the value $+infty$ for every initial choices of $mu,nu,gamma$, if we exclude the case where $gamma$ has marginals $mu$ and $nu$?
real-analysis analysis probability-theory optimization supremum-and-infimum
asked Mar 12 at 22:09
user63841219user63841219
1468
$endgroup$
add a comment |
$begingroup$
Let $X, Y$ be Polish spaces with probability measures $mu, nu$, respectively; and let $gamma$ be a finite measure on the product $X times Y$. We consider the maximization problem
$$
sup_{varphi,psi} bigg{int_{X} varphi dmu + int_{Y} psi dnu - int_{X times Y} big( varphi(x) + psi(y) big) dgammabigg}
$$
over the set of all bounded and continuous functions $varphi(x),psi(y)$.
Why can this maximization problem always attain the value $+infty$ for every initial choices of $mu,nu,gamma$, if we exclude the case where $gamma$ has marginals $mu$ and $nu$?
real-analysis analysis probability-theory optimization supremum-and-infimum
asked Mar 12 at 22:09
user63841219user63841219
1468
$endgroup$
$begingroup$
If $gamma$ required to be a probability measure?
$endgroup$
– kimchi lover
Mar 12 at 22:32
$begingroup$
$gamma$ is a finite measure on $X times Y$.
$endgroup$
– user63841219
Mar 12 at 22:42
add a comment |
$begingroup$
Let $X, Y$ be Polish spaces with probability measures $mu, nu$, respectively; and let $gamma$ be a finite measure on the product $X times Y$. We consider the maximization problem
$$
sup_{varphi,psi} bigg{int_{X} varphi dmu + int_{Y} psi dnu - int_{X times Y} big( varphi(x) + psi(y) big) dgammabigg}
$$
over the set of all bounded and continuous functions $varphi(x),psi(y)$.
Why can this maximization problem always attain the value $+infty$ for every initial choices of $mu,nu,gamma$, if we exclude the case where $gamma$ has marginals $mu$ and $nu$?
real-analysis analysis probability-theory optimization supremum-and-infimum
asked Mar 12 at 22:09
user63841219user63841219
1468
$endgroup$
Let $X, Y$ be Polish spaces with probability measures $mu, nu$, respectively; and let $gamma$ be a finite measure on the product $X times Y$. We consider the maximization problem
$$
sup_{varphi,psi} bigg{int_{X} varphi dmu + int_{Y} psi dnu - int_{X times Y} big( varphi(x) + psi(y) big) dgammabigg}
$$
over the set of all bounded and continuous functions $varphi(x),psi(y)$.
Why can this maximization problem always attain the value $+infty$ for every initial choices of $mu,nu,gamma$, if we exclude the case where $gamma$ has marginals $mu$ and $nu$?
real-analysis analysis probability-theory optimization supremum-and-infimum
real-analysis analysis probability-theory optimization supremum-and-infimum
asked Mar 12 at 22:09
user63841219user63841219
1468
asked Mar 12 at 22:09
user63841219user63841219
1468
edited Mar 12 at 22:41
user63841219
asked Mar 12 at 22:09
user63841219user63841219
1468
asked Mar 12 at 22:09
user63841219user63841219
1468
asked Mar 12 at 22:09
user63841219user63841219
1468
1468
$begingroup$
If $gamma$ required to be a probability measure?
$endgroup$
– kimchi lover
Mar 12 at 22:32
$begingroup$
$gamma$ is a finite measure on $X times Y$.
$endgroup$
– user63841219
Mar 12 at 22:42
add a comment |
$begingroup$
If $gamma$ required to be a probability measure?
$endgroup$
– kimchi lover
Mar 12 at 22:32
$begingroup$
$gamma$ is a finite measure on $X times Y$.
$endgroup$
– user63841219
Mar 12 at 22:42
$begingroup$
If $gamma$ required to be a probability measure?
$endgroup$
– kimchi lover
Mar 12 at 22:32
$begingroup$
If $gamma$ required to be a probability measure?
$endgroup$
– kimchi lover
Mar 12 at 22:32
$begingroup$
$gamma$ is a finite measure on $X times Y$.
$endgroup$
– user63841219
Mar 12 at 22:42
$begingroup$
$gamma$ is a finite measure on $X times Y$.
$endgroup$
– user63841219
Mar 12 at 22:42
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $mu'$ and $nu'$ be the margins of $gamma$. At least one of $munemu'$ and $nunenu'$ must hold.
Your maximization problem becomes: maximize $$int_X varphi(dmu-dmu') +int_Ypsi(dnu-dnu').$$
Assume without loss of generality that $munemu'$, let the signed measure $sigma=mu-mu'.$ Then your maximization problem is lower bounded by the restricted maximization problem, of determining $$sup_varphi int_{X}varphi dsigma.$$ Since $sigma$ is not the zero measure, there exists a continuous bounded $f$ such that $int_X fsigma = 1$. Now it is clear that by taking $varphi$ to be large scalar multiples of $f$, you see the supremum is $infty$.
answered Mar 12 at 23:08
kimchi loverkimchi lover
11.3k31229
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Let $mu'$ and $nu'$ be the margins of $gamma$. At least one of $munemu'$ and $nunenu'$ must hold.
Your maximization problem becomes: maximize $$int_X varphi(dmu-dmu') +int_Ypsi(dnu-dnu').$$
Assume without loss of generality that $munemu'$, let the signed measure $sigma=mu-mu'.$ Then your maximization problem is lower bounded by the restricted maximization problem, of determining $$sup_varphi int_{X}varphi dsigma.$$ Since $sigma$ is not the zero measure, there exists a continuous bounded $f$ such that $int_X fsigma = 1$. Now it is clear that by taking $varphi$ to be large scalar multiples of $f$, you see the supremum is $infty$.
answered Mar 12 at 23:08
kimchi loverkimchi lover
11.3k31229
$endgroup$
add a comment |
$begingroup$
Let $mu'$ and $nu'$ be the margins of $gamma$. At least one of $munemu'$ and $nunenu'$ must hold.
Your maximization problem becomes: maximize $$int_X varphi(dmu-dmu') +int_Ypsi(dnu-dnu').$$
Assume without loss of generality that $munemu'$, let the signed measure $sigma=mu-mu'.$ Then your maximization problem is lower bounded by the restricted maximization problem, of determining $$sup_varphi int_{X}varphi dsigma.$$ Since $sigma$ is not the zero measure, there exists a continuous bounded $f$ such that $int_X fsigma = 1$. Now it is clear that by taking $varphi$ to be large scalar multiples of $f$, you see the supremum is $infty$.
answered Mar 12 at 23:08
kimchi loverkimchi lover
11.3k31229
$endgroup$
add a comment |
$begingroup$
Let $mu'$ and $nu'$ be the margins of $gamma$. At least one of $munemu'$ and $nunenu'$ must hold.
Your maximization problem becomes: maximize $$int_X varphi(dmu-dmu') +int_Ypsi(dnu-dnu').$$
Assume without loss of generality that $munemu'$, let the signed measure $sigma=mu-mu'.$ Then your maximization problem is lower bounded by the restricted maximization problem, of determining $$sup_varphi int_{X}varphi dsigma.$$ Since $sigma$ is not the zero measure, there exists a continuous bounded $f$ such that $int_X fsigma = 1$. Now it is clear that by taking $varphi$ to be large scalar multiples of $f$, you see the supremum is $infty$.
answered Mar 12 at 23:08
kimchi loverkimchi lover
11.3k31229
$endgroup$
Let $mu'$ and $nu'$ be the margins of $gamma$. At least one of $munemu'$ and $nunenu'$ must hold.
Your maximization problem becomes: maximize $$int_X varphi(dmu-dmu') +int_Ypsi(dnu-dnu').$$
Assume without loss of generality that $munemu'$, let the signed measure $sigma=mu-mu'.$ Then your maximization problem is lower bounded by the restricted maximization problem, of determining $$sup_varphi int_{X}varphi dsigma.$$ Since $sigma$ is not the zero measure, there exists a continuous bounded $f$ such that $int_X fsigma = 1$. Now it is clear that by taking $varphi$ to be large scalar multiples of $f$, you see the supremum is $infty$.
answered Mar 12 at 23:08
kimchi loverkimchi lover
11.3k31229
answered Mar 12 at 23:08
kimchi loverkimchi lover
11.3k31229
answered Mar 12 at 23:08
kimchi loverkimchi lover
11.3k31229
answered Mar 12 at 23:08
kimchi loverkimchi lover
11.3k31229
11.3k31229
add a comment |
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$begingroup$
If $gamma$ required to be a probability measure?
$endgroup$
– kimchi lover
Mar 12 at 22:32
$begingroup$
$gamma$ is a finite measure on $X times Y$.
$endgroup$
– user63841219
Mar 12 at 22:42