Mixing
Equiintegrability of some family of sequences
$begingroup$
Let $(rho_n)_nsubset L^1$ be a Dirac sequence. Study the equiintegrability of the following family:
(a) $f_n=rho_n^2$,
(b) $g_n=rho_1*rho_n$ (convolution),
(c) $g_n=rho_1*rho_n^2$.
My attempt
Since for any $epsilon>0$ we have $int_{|x|<epsilon}|rho_n(x)|dxrightarrow 1$, $(rho_n)_n$ is not equiintegrable.
(b) Let $epsilon>0$. Choose $delta$ such that $m(A)leq delta$ implies $int_A|rho_1(x)|dx<epsilon$. Then
begin{align*} int_A|g_n(x)|dx&=int_A|rho_1*rho_n|dx\
&=int_Aint_{mathbb{R}}|rho_1(x-y)| |rho_n(y)|dydx\
&=int_{mathbb{R}} |rho_n(y)|left(int_A|rho_1(x-y)|dxright)dy\
&leq epsilon int_{mathbb{R}} |rho_n(y)|dy=epsilon.
end{align*}
Hence, the family $(g_n)_n$ is equiintegrable.
(a) I suspect its not equintegrable but I do not know what procee.
begin{align*} int_A|f_n(x)|dx&=int_A|rho_n^2|dx.
end{align*}
I hope to get this kind of inequality
begin{align*} int_A|rho_n^2|dx&leqint_A|rho_n|dx int_A|rho_n|dx.
end{align*}
functional-analysis measure-theory convergence lebesgue-integral lp-spaces
asked Jan 31 at 15:47
Muhammad MubarakMuhammad Mubarak
9810
$endgroup$
add a comment |
$begingroup$
Let $(rho_n)_nsubset L^1$ be a Dirac sequence. Study the equiintegrability of the following family:
(a) $f_n=rho_n^2$,
(b) $g_n=rho_1*rho_n$ (convolution),
(c) $g_n=rho_1*rho_n^2$.
My attempt
Since for any $epsilon>0$ we have $int_{|x|<epsilon}|rho_n(x)|dxrightarrow 1$, $(rho_n)_n$ is not equiintegrable.
(b) Let $epsilon>0$. Choose $delta$ such that $m(A)leq delta$ implies $int_A|rho_1(x)|dx<epsilon$. Then
begin{align*} int_A|g_n(x)|dx&=int_A|rho_1*rho_n|dx\
&=int_Aint_{mathbb{R}}|rho_1(x-y)| |rho_n(y)|dydx\
&=int_{mathbb{R}} |rho_n(y)|left(int_A|rho_1(x-y)|dxright)dy\
&leq epsilon int_{mathbb{R}} |rho_n(y)|dy=epsilon.
end{align*}
Hence, the family $(g_n)_n$ is equiintegrable.
(a) I suspect its not equintegrable but I do not know what procee.
begin{align*} int_A|f_n(x)|dx&=int_A|rho_n^2|dx.
end{align*}
I hope to get this kind of inequality
begin{align*} int_A|rho_n^2|dx&leqint_A|rho_n|dx int_A|rho_n|dx.
end{align*}
functional-analysis measure-theory convergence lebesgue-integral lp-spaces
asked Jan 31 at 15:47
Muhammad MubarakMuhammad Mubarak
9810
$endgroup$
$begingroup$
What is a Dirac sequence and are you sure it is in $mathbf{R}$? It sounds like the $rho_n$ are measures.
$endgroup$
– Will M.
Jan 31 at 18:07
$begingroup$
A Dirac sequence is one that satisfies $rho_ngeq 0$, $int_Rrho=1$ and $int_{|x|>epsilon}rho_nrightarrow 0$. I mean in that that the domain is $R$
$endgroup$
– Muhammad Mubarak
Jan 31 at 18:29
add a comment |
$begingroup$
Let $(rho_n)_nsubset L^1$ be a Dirac sequence. Study the equiintegrability of the following family:
(a) $f_n=rho_n^2$,
(b) $g_n=rho_1*rho_n$ (convolution),
(c) $g_n=rho_1*rho_n^2$.
My attempt
Since for any $epsilon>0$ we have $int_{|x|<epsilon}|rho_n(x)|dxrightarrow 1$, $(rho_n)_n$ is not equiintegrable.
(b) Let $epsilon>0$. Choose $delta$ such that $m(A)leq delta$ implies $int_A|rho_1(x)|dx<epsilon$. Then
begin{align*} int_A|g_n(x)|dx&=int_A|rho_1*rho_n|dx\
&=int_Aint_{mathbb{R}}|rho_1(x-y)| |rho_n(y)|dydx\
&=int_{mathbb{R}} |rho_n(y)|left(int_A|rho_1(x-y)|dxright)dy\
&leq epsilon int_{mathbb{R}} |rho_n(y)|dy=epsilon.
end{align*}
Hence, the family $(g_n)_n$ is equiintegrable.
(a) I suspect its not equintegrable but I do not know what procee.
begin{align*} int_A|f_n(x)|dx&=int_A|rho_n^2|dx.
end{align*}
I hope to get this kind of inequality
begin{align*} int_A|rho_n^2|dx&leqint_A|rho_n|dx int_A|rho_n|dx.
end{align*}
functional-analysis measure-theory convergence lebesgue-integral lp-spaces
asked Jan 31 at 15:47
Muhammad MubarakMuhammad Mubarak
9810
$endgroup$
Let $(rho_n)_nsubset L^1$ be a Dirac sequence. Study the equiintegrability of the following family:
(a) $f_n=rho_n^2$,
(b) $g_n=rho_1*rho_n$ (convolution),
(c) $g_n=rho_1*rho_n^2$.
My attempt
Since for any $epsilon>0$ we have $int_{|x|<epsilon}|rho_n(x)|dxrightarrow 1$, $(rho_n)_n$ is not equiintegrable.
(b) Let $epsilon>0$. Choose $delta$ such that $m(A)leq delta$ implies $int_A|rho_1(x)|dx<epsilon$. Then
begin{align*} int_A|g_n(x)|dx&=int_A|rho_1*rho_n|dx\
&=int_Aint_{mathbb{R}}|rho_1(x-y)| |rho_n(y)|dydx\
&=int_{mathbb{R}} |rho_n(y)|left(int_A|rho_1(x-y)|dxright)dy\
&leq epsilon int_{mathbb{R}} |rho_n(y)|dy=epsilon.
end{align*}
Hence, the family $(g_n)_n$ is equiintegrable.
(a) I suspect its not equintegrable but I do not know what procee.
begin{align*} int_A|f_n(x)|dx&=int_A|rho_n^2|dx.
end{align*}
I hope to get this kind of inequality
begin{align*} int_A|rho_n^2|dx&leqint_A|rho_n|dx int_A|rho_n|dx.
end{align*}
functional-analysis measure-theory convergence lebesgue-integral lp-spaces
functional-analysis measure-theory convergence lebesgue-integral lp-spaces
asked Jan 31 at 15:47
Muhammad MubarakMuhammad Mubarak
9810
asked Jan 31 at 15:47
Muhammad MubarakMuhammad Mubarak
9810
edited Jan 31 at 18:30
Muhammad Mubarak
asked Jan 31 at 15:47
Muhammad MubarakMuhammad Mubarak
9810
asked Jan 31 at 15:47
Muhammad MubarakMuhammad Mubarak
9810
asked Jan 31 at 15:47
Muhammad MubarakMuhammad Mubarak
9810
9810
$begingroup$
What is a Dirac sequence and are you sure it is in $mathbf{R}$? It sounds like the $rho_n$ are measures.
$endgroup$
– Will M.
Jan 31 at 18:07
$begingroup$
A Dirac sequence is one that satisfies $rho_ngeq 0$, $int_Rrho=1$ and $int_{|x|>epsilon}rho_nrightarrow 0$. I mean in that that the domain is $R$
$endgroup$
– Muhammad Mubarak
Jan 31 at 18:29
add a comment |
$begingroup$
What is a Dirac sequence and are you sure it is in $mathbf{R}$? It sounds like the $rho_n$ are measures.
$endgroup$
– Will M.
Jan 31 at 18:07
$begingroup$
A Dirac sequence is one that satisfies $rho_ngeq 0$, $int_Rrho=1$ and $int_{|x|>epsilon}rho_nrightarrow 0$. I mean in that that the domain is $R$
$endgroup$
– Muhammad Mubarak
Jan 31 at 18:29
$begingroup$
What is a Dirac sequence and are you sure it is in $mathbf{R}$? It sounds like the $rho_n$ are measures.
$endgroup$
– Will M.
Jan 31 at 18:07
$begingroup$
What is a Dirac sequence and are you sure it is in $mathbf{R}$? It sounds like the $rho_n$ are measures.
$endgroup$
– Will M.
Jan 31 at 18:07
$begingroup$
A Dirac sequence is one that satisfies $rho_ngeq 0$, $int_Rrho=1$ and $int_{|x|>epsilon}rho_nrightarrow 0$. I mean in that that the domain is $R$
$endgroup$
– Muhammad Mubarak
Jan 31 at 18:29
$begingroup$
A Dirac sequence is one that satisfies $rho_ngeq 0$, $int_Rrho=1$ and $int_{|x|>epsilon}rho_nrightarrow 0$. I mean in that that the domain is $R$
$endgroup$
– Muhammad Mubarak
Jan 31 at 18:29
add a comment |
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$begingroup$
What is a Dirac sequence and are you sure it is in $mathbf{R}$? It sounds like the $rho_n$ are measures.
$endgroup$
– Will M.
Jan 31 at 18:07
$begingroup$
A Dirac sequence is one that satisfies $rho_ngeq 0$, $int_Rrho=1$ and $int_{|x|>epsilon}rho_nrightarrow 0$. I mean in that that the domain is $R$
$endgroup$
– Muhammad Mubarak
Jan 31 at 18:29