Let $f:(0;infty)times (0;infty) rightarrow Bbb R$ be a Lebesgue measurable function, is...
$begingroup$
Let $f:(0;infty)times (0;infty) rightarrow Bbb R$ be a Lebesgue measurable function. Lets define
$$T_f(y)=int_0^{+infty} f(x,y) , dy$$
$$G_f(y)=int_0^{+infty} |f(x,y)| , dy$$
By Tonelli's Theorem I know $G_f$ is measurable and I also know $G_f in L^p(0;+infty)$ (take it as an hypothesis). I would like to say $Vert T_f Vert _p le Vert G_f Vert _p$ so $T_f in L^p(0;+infty)$ but I need $T_f$ to be measurable in order to be abel to calculate $Vert T_f Vert _p$. Is this true? How can I prove it?
real-analysis analysis lebesgue-integral lebesgue-measure
asked Dec 15 '18 at 23:00
Marcos Martínez WagnerMarcos Martínez Wagner
1148
$endgroup$
|
show 3 more comments
$begingroup$
Let $f:(0;infty)times (0;infty) rightarrow Bbb R$ be a Lebesgue measurable function. Lets define
$$T_f(y)=int_0^{+infty} f(x,y) , dy$$
$$G_f(y)=int_0^{+infty} |f(x,y)| , dy$$
By Tonelli's Theorem I know $G_f$ is measurable and I also know $G_f in L^p(0;+infty)$ (take it as an hypothesis). I would like to say $Vert T_f Vert _p le Vert G_f Vert _p$ so $T_f in L^p(0;+infty)$ but I need $T_f$ to be measurable in order to be abel to calculate $Vert T_f Vert _p$. Is this true? How can I prove it?
real-analysis analysis lebesgue-integral lebesgue-measure
asked Dec 15 '18 at 23:00
Marcos Martínez WagnerMarcos Martínez Wagner
1148
$endgroup$
$begingroup$
$T_f$ is a pointwise limit of measurable functions (truncate the integral at $n$, then you can use the full Fubini theorem)
$endgroup$
– Mindlack
Dec 15 '18 at 23:03
$begingroup$
wich is the difference between "full Fubini Theorem" and "Fubini Theorem"?
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:11
$begingroup$
To me Fubini-Tonelli refers to the Fubini theorem when everything is non-negative (which is easier). By “full Fubini” I thought of the theorem when your function is not non-negative. So nothing really important.
$endgroup$
– Mindlack
Dec 15 '18 at 23:13
$begingroup$
Thank you!! I think I got it
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:19
$begingroup$
Nol, in fact the integral defining $T_f$ need not exist. (Yes if $f$ is integrable on the product space.)
$endgroup$
– David C. Ullrich
Dec 15 '18 at 23:22
|
show 3 more comments
$begingroup$
Let $f:(0;infty)times (0;infty) rightarrow Bbb R$ be a Lebesgue measurable function. Lets define
$$T_f(y)=int_0^{+infty} f(x,y) , dy$$
$$G_f(y)=int_0^{+infty} |f(x,y)| , dy$$
By Tonelli's Theorem I know $G_f$ is measurable and I also know $G_f in L^p(0;+infty)$ (take it as an hypothesis). I would like to say $Vert T_f Vert _p le Vert G_f Vert _p$ so $T_f in L^p(0;+infty)$ but I need $T_f$ to be measurable in order to be abel to calculate $Vert T_f Vert _p$. Is this true? How can I prove it?
real-analysis analysis lebesgue-integral lebesgue-measure
asked Dec 15 '18 at 23:00
Marcos Martínez WagnerMarcos Martínez Wagner
1148
$endgroup$
Let $f:(0;infty)times (0;infty) rightarrow Bbb R$ be a Lebesgue measurable function. Lets define
$$T_f(y)=int_0^{+infty} f(x,y) , dy$$
$$G_f(y)=int_0^{+infty} |f(x,y)| , dy$$
By Tonelli's Theorem I know $G_f$ is measurable and I also know $G_f in L^p(0;+infty)$ (take it as an hypothesis). I would like to say $Vert T_f Vert _p le Vert G_f Vert _p$ so $T_f in L^p(0;+infty)$ but I need $T_f$ to be measurable in order to be abel to calculate $Vert T_f Vert _p$. Is this true? How can I prove it?
real-analysis analysis lebesgue-integral lebesgue-measure
real-analysis analysis lebesgue-integral lebesgue-measure
asked Dec 15 '18 at 23:00
Marcos Martínez WagnerMarcos Martínez Wagner
1148
asked Dec 15 '18 at 23:00
Marcos Martínez WagnerMarcos Martínez Wagner
1148
asked Dec 15 '18 at 23:00
Marcos Martínez WagnerMarcos Martínez Wagner
1148
asked Dec 15 '18 at 23:00
Marcos Martínez WagnerMarcos Martínez Wagner
1148
asked Dec 15 '18 at 23:00
Marcos Martínez WagnerMarcos Martínez Wagner
1148
1148
$begingroup$
$T_f$ is a pointwise limit of measurable functions (truncate the integral at $n$, then you can use the full Fubini theorem)
$endgroup$
– Mindlack
Dec 15 '18 at 23:03
$begingroup$
wich is the difference between "full Fubini Theorem" and "Fubini Theorem"?
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:11
$begingroup$
To me Fubini-Tonelli refers to the Fubini theorem when everything is non-negative (which is easier). By “full Fubini” I thought of the theorem when your function is not non-negative. So nothing really important.
$endgroup$
– Mindlack
Dec 15 '18 at 23:13
$begingroup$
Thank you!! I think I got it
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:19
$begingroup$
Nol, in fact the integral defining $T_f$ need not exist. (Yes if $f$ is integrable on the product space.)
$endgroup$
– David C. Ullrich
Dec 15 '18 at 23:22
|
show 3 more comments
$begingroup$
$T_f$ is a pointwise limit of measurable functions (truncate the integral at $n$, then you can use the full Fubini theorem)
$endgroup$
– Mindlack
Dec 15 '18 at 23:03
$begingroup$
wich is the difference between "full Fubini Theorem" and "Fubini Theorem"?
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:11
$begingroup$
To me Fubini-Tonelli refers to the Fubini theorem when everything is non-negative (which is easier). By “full Fubini” I thought of the theorem when your function is not non-negative. So nothing really important.
$endgroup$
– Mindlack
Dec 15 '18 at 23:13
$begingroup$
Thank you!! I think I got it
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:19
$begingroup$
Nol, in fact the integral defining $T_f$ need not exist. (Yes if $f$ is integrable on the product space.)
$endgroup$
– David C. Ullrich
Dec 15 '18 at 23:22
$begingroup$
$T_f$ is a pointwise limit of measurable functions (truncate the integral at $n$, then you can use the full Fubini theorem)
$endgroup$
– Mindlack
Dec 15 '18 at 23:03
$begingroup$
$T_f$ is a pointwise limit of measurable functions (truncate the integral at $n$, then you can use the full Fubini theorem)
$endgroup$
– Mindlack
Dec 15 '18 at 23:03
$begingroup$
wich is the difference between "full Fubini Theorem" and "Fubini Theorem"?
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:11
$begingroup$
wich is the difference between "full Fubini Theorem" and "Fubini Theorem"?
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:11
$begingroup$
To me Fubini-Tonelli refers to the Fubini theorem when everything is non-negative (which is easier). By “full Fubini” I thought of the theorem when your function is not non-negative. So nothing really important.
$endgroup$
– Mindlack
Dec 15 '18 at 23:13
$begingroup$
To me Fubini-Tonelli refers to the Fubini theorem when everything is non-negative (which is easier). By “full Fubini” I thought of the theorem when your function is not non-negative. So nothing really important.
$endgroup$
– Mindlack
Dec 15 '18 at 23:13
$begingroup$
Thank you!! I think I got it
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:19
$begingroup$
Thank you!! I think I got it
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:19
$begingroup$
Nol, in fact the integral defining $T_f$ need not exist. (Yes if $f$ is integrable on the product space.)
$endgroup$
– David C. Ullrich
Dec 15 '18 at 23:22
$begingroup$
Nol, in fact the integral defining $T_f$ need not exist. (Yes if $f$ is integrable on the product space.)
$endgroup$
– David C. Ullrich
Dec 15 '18 at 23:22
|
show 3 more comments
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$begingroup$
$T_f$ is a pointwise limit of measurable functions (truncate the integral at $n$, then you can use the full Fubini theorem)
$endgroup$
– Mindlack
Dec 15 '18 at 23:03
$begingroup$
wich is the difference between "full Fubini Theorem" and "Fubini Theorem"?
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:11
$begingroup$
To me Fubini-Tonelli refers to the Fubini theorem when everything is non-negative (which is easier). By “full Fubini” I thought of the theorem when your function is not non-negative. So nothing really important.
$endgroup$
– Mindlack
Dec 15 '18 at 23:13
$begingroup$
Thank you!! I think I got it
$endgroup$
– Marcos Martínez Wagner
Dec 15 '18 at 23:19
$begingroup$
Nol, in fact the integral defining $T_f$ need not exist. (Yes if $f$ is integrable on the product space.)
$endgroup$
– David C. Ullrich
Dec 15 '18 at 23:22