Exchange supremum parameter dependent and integral
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Suppose that on a complete and separable metric space $(X,d)$ we have a function $f:X to [0,+infty]$ which is Borel. Consider for every $tin [0,1]$ the evaluation map $e_t:C([0,1],X) to X$. and $(pi_n)_n$ a sequence of Borel probability measure on the space of curves. Suppose that
$$
sup_n in mathbbNsup_tin [0,1] int f(e_t) mathrmdpi_n < C
$$
for some constant $Cin mathbbR$. In which case I can say that $sup_t f(e_t)$ is measurable and that
$$
sup_n in mathbbN int sup_tin [0,1] f(e_t) mathrmdpi_n < Cqquad ?
$$
Thank you.
real-analysis measure-theory metric-spaces supremum-and-infimum
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up vote
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down vote
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Suppose that on a complete and separable metric space $(X,d)$ we have a function $f:X to [0,+infty]$ which is Borel. Consider for every $tin [0,1]$ the evaluation map $e_t:C([0,1],X) to X$. and $(pi_n)_n$ a sequence of Borel probability measure on the space of curves. Suppose that
$$
sup_n in mathbbNsup_tin [0,1] int f(e_t) mathrmdpi_n < C
$$
for some constant $Cin mathbbR$. In which case I can say that $sup_t f(e_t)$ is measurable and that
$$
sup_n in mathbbN int sup_tin [0,1] f(e_t) mathrmdpi_n < Cqquad ?
$$
Thank you.
real-analysis measure-theory metric-spaces supremum-and-infimum
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose that on a complete and separable metric space $(X,d)$ we have a function $f:X to [0,+infty]$ which is Borel. Consider for every $tin [0,1]$ the evaluation map $e_t:C([0,1],X) to X$. and $(pi_n)_n$ a sequence of Borel probability measure on the space of curves. Suppose that
$$
sup_n in mathbbNsup_tin [0,1] int f(e_t) mathrmdpi_n < C
$$
for some constant $Cin mathbbR$. In which case I can say that $sup_t f(e_t)$ is measurable and that
$$
sup_n in mathbbN int sup_tin [0,1] f(e_t) mathrmdpi_n < Cqquad ?
$$
Thank you.
real-analysis measure-theory metric-spaces supremum-and-infimum
Suppose that on a complete and separable metric space $(X,d)$ we have a function $f:X to [0,+infty]$ which is Borel. Consider for every $tin [0,1]$ the evaluation map $e_t:C([0,1],X) to X$. and $(pi_n)_n$ a sequence of Borel probability measure on the space of curves. Suppose that
$$
sup_n in mathbbNsup_tin [0,1] int f(e_t) mathrmdpi_n < C
$$
for some constant $Cin mathbbR$. In which case I can say that $sup_t f(e_t)$ is measurable and that
$$
sup_n in mathbbN int sup_tin [0,1] f(e_t) mathrmdpi_n < Cqquad ?
$$
Thank you.
real-analysis measure-theory metric-spaces supremum-and-infimum
asked Aug 8 at 20:43
Luca Benatti
385
385
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