Some sort of Laplace transform
$begingroup$
Let $K$ be some compact set and $mu$ be a probability measure on $K$.
Let $f:Ktomathbb S^{d-1}$ be a continuous function, where $mathbb S^{d-1}$ is the unit sphere in $mathbb R^d$ And $g:Ktomathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $yinmathbb R^d$,
$$int_K e^{y^{top} f(x)}y^top g(x) dmu(x)=0,$$
Then $g(x)=0$ $mu$-almost surely?
When $mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case...
Thank you!
real-analysis laplace-transform
asked Jan 5 at 18:39
TrivialPursuitTrivialPursuit
63
$endgroup$
add a comment |
$begingroup$
Let $K$ be some compact set and $mu$ be a probability measure on $K$.
Let $f:Ktomathbb S^{d-1}$ be a continuous function, where $mathbb S^{d-1}$ is the unit sphere in $mathbb R^d$ And $g:Ktomathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $yinmathbb R^d$,
$$int_K e^{y^{top} f(x)}y^top g(x) dmu(x)=0,$$
Then $g(x)=0$ $mu$-almost surely?
When $mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case...
Thank you!
real-analysis laplace-transform
asked Jan 5 at 18:39
TrivialPursuitTrivialPursuit
63
$endgroup$
add a comment |
$begingroup$
Let $K$ be some compact set and $mu$ be a probability measure on $K$.
Let $f:Ktomathbb S^{d-1}$ be a continuous function, where $mathbb S^{d-1}$ is the unit sphere in $mathbb R^d$ And $g:Ktomathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $yinmathbb R^d$,
$$int_K e^{y^{top} f(x)}y^top g(x) dmu(x)=0,$$
Then $g(x)=0$ $mu$-almost surely?
When $mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case...
Thank you!
real-analysis laplace-transform
asked Jan 5 at 18:39
TrivialPursuitTrivialPursuit
63
$endgroup$
Let $K$ be some compact set and $mu$ be a probability measure on $K$.
Let $f:Ktomathbb S^{d-1}$ be a continuous function, where $mathbb S^{d-1}$ is the unit sphere in $mathbb R^d$ And $g:Ktomathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $yinmathbb R^d$,
$$int_K e^{y^{top} f(x)}y^top g(x) dmu(x)=0,$$
Then $g(x)=0$ $mu$-almost surely?
When $mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case...
Thank you!
real-analysis laplace-transform
real-analysis laplace-transform
asked Jan 5 at 18:39
TrivialPursuitTrivialPursuit
63
asked Jan 5 at 18:39
TrivialPursuitTrivialPursuit
63
asked Jan 5 at 18:39
TrivialPursuitTrivialPursuit
63
asked Jan 5 at 18:39
TrivialPursuitTrivialPursuit
63
asked Jan 5 at 18:39
TrivialPursuitTrivialPursuit
63
63
add a comment |
add a comment |
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