Function continuous in one variable, Borel in other is jointly Borel?
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Suppose I have a function $f: Xtimes Yto mathbb{C}$, where $X$ and $Y$ are compact Hausdorff spaces, where the function is continuous in $X$ and Borel in $Y$. Is it true that the function is jointly Borel in $Xtimes Y$? What additional conditions might ensure this?
Thanks.
real-analysis functional-analysis measure-theory
asked Nov 16 at 23:18
Merry
398110
add a comment |
up vote
1
down vote
favorite
Suppose I have a function $f: Xtimes Yto mathbb{C}$, where $X$ and $Y$ are compact Hausdorff spaces, where the function is continuous in $X$ and Borel in $Y$. Is it true that the function is jointly Borel in $Xtimes Y$? What additional conditions might ensure this?
Thanks.
real-analysis functional-analysis measure-theory
asked Nov 16 at 23:18
Merry
398110
According to encyclopediaofmath.org/index.php/Measurable_space it holds if $X$ is second countable.
– MaoWao
Nov 16 at 23:56
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose I have a function $f: Xtimes Yto mathbb{C}$, where $X$ and $Y$ are compact Hausdorff spaces, where the function is continuous in $X$ and Borel in $Y$. Is it true that the function is jointly Borel in $Xtimes Y$? What additional conditions might ensure this?
Thanks.
real-analysis functional-analysis measure-theory
asked Nov 16 at 23:18
Merry
398110
Suppose I have a function $f: Xtimes Yto mathbb{C}$, where $X$ and $Y$ are compact Hausdorff spaces, where the function is continuous in $X$ and Borel in $Y$. Is it true that the function is jointly Borel in $Xtimes Y$? What additional conditions might ensure this?
Thanks.
real-analysis functional-analysis measure-theory
real-analysis functional-analysis measure-theory
asked Nov 16 at 23:18
Merry
398110
asked Nov 16 at 23:18
Merry
398110
edited Nov 16 at 23:38
asked Nov 16 at 23:18
Merry
398110
asked Nov 16 at 23:18
Merry
398110
asked Nov 16 at 23:18
Merry
398110
398110
According to encyclopediaofmath.org/index.php/Measurable_space it holds if $X$ is second countable.
– MaoWao
Nov 16 at 23:56
add a comment |
According to encyclopediaofmath.org/index.php/Measurable_space it holds if $X$ is second countable.
– MaoWao
Nov 16 at 23:56
According to encyclopediaofmath.org/index.php/Measurable_space it holds if $X$ is second countable.
– MaoWao
Nov 16 at 23:56
According to encyclopediaofmath.org/index.php/Measurable_space it holds if $X$ is second countable.
– MaoWao
Nov 16 at 23:56
add a comment |
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According to encyclopediaofmath.org/index.php/Measurable_space it holds if $X$ is second countable.
– MaoWao
Nov 16 at 23:56