Is there a well defined convolution where Fubini's theorem fails?
$begingroup$
Suppose $f$ and $g$ are real functions such that $f*gin L^1$ Is there an example where Fubini's theorem might fail for the integral of the convolution? That is:
$$int_X(int_Yf(x-y)g(y)dy)dxnot=int_Y(int_Xf(x-y)g(y)dx)dy$$
I know that if $f,gin L^1$ then by Tonelli the product $f(x-y)g(y)$ is integrable in $Xtimes Y$ and hence Fubini's theorem holds, but is there an example where that's not the case?
lebesgue-integral convolution iterated-integrals
asked Dec 31 '18 at 21:46
Bar AlonBar Alon
529217
$endgroup$
add a comment |
$begingroup$
Suppose $f$ and $g$ are real functions such that $f*gin L^1$ Is there an example where Fubini's theorem might fail for the integral of the convolution? That is:
$$int_X(int_Yf(x-y)g(y)dy)dxnot=int_Y(int_Xf(x-y)g(y)dx)dy$$
I know that if $f,gin L^1$ then by Tonelli the product $f(x-y)g(y)$ is integrable in $Xtimes Y$ and hence Fubini's theorem holds, but is there an example where that's not the case?
lebesgue-integral convolution iterated-integrals
asked Dec 31 '18 at 21:46
Bar AlonBar Alon
529217
$endgroup$
add a comment |
$begingroup$
Suppose $f$ and $g$ are real functions such that $f*gin L^1$ Is there an example where Fubini's theorem might fail for the integral of the convolution? That is:
$$int_X(int_Yf(x-y)g(y)dy)dxnot=int_Y(int_Xf(x-y)g(y)dx)dy$$
I know that if $f,gin L^1$ then by Tonelli the product $f(x-y)g(y)$ is integrable in $Xtimes Y$ and hence Fubini's theorem holds, but is there an example where that's not the case?
lebesgue-integral convolution iterated-integrals
asked Dec 31 '18 at 21:46
Bar AlonBar Alon
529217
$endgroup$
Suppose $f$ and $g$ are real functions such that $f*gin L^1$ Is there an example where Fubini's theorem might fail for the integral of the convolution? That is:
$$int_X(int_Yf(x-y)g(y)dy)dxnot=int_Y(int_Xf(x-y)g(y)dx)dy$$
I know that if $f,gin L^1$ then by Tonelli the product $f(x-y)g(y)$ is integrable in $Xtimes Y$ and hence Fubini's theorem holds, but is there an example where that's not the case?
lebesgue-integral convolution iterated-integrals
lebesgue-integral convolution iterated-integrals
asked Dec 31 '18 at 21:46
Bar AlonBar Alon
529217
asked Dec 31 '18 at 21:46
Bar AlonBar Alon
529217
asked Dec 31 '18 at 21:46
Bar AlonBar Alon
529217
asked Dec 31 '18 at 21:46
Bar AlonBar Alon
529217
asked Dec 31 '18 at 21:46
Bar AlonBar Alon
529217
529217
add a comment |
add a comment |
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