Ask a question about measurable function












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Suppose $f(x,y)$ is a function on $mathbb{R}^2$ such that:



1) For a fixed $x$, $f(x,y)$ is a measurable function of $y$ and
2) For a fixed $y$, $f(x,y)$ is a continuous function of $x$.



Prove that for a function $g(y)$ that is measurable on $mathbb{R}$, we have that $f(g(y),y)$ is measurable on $mathbb{R}$.



I can prove that f(x,y)is measurable on $mathbb{R}^2$,but I don't know how to prove this problem?



For f(x,y) is just measurable on $mathbb{R}^2$,is the result still right?










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  • $begingroup$
    Welcome to MSE. Please read this text about how to ask a good question.
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    – José Carlos Santos
    Dec 7 '18 at 9:37










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    You say that you know how to prove that $f(x,y)$ is measurable. Also $ymapsto (g(y),y)$ is measurable, right? What happens when you take the composition?
    $endgroup$
    – Federico
    Dec 7 '18 at 18:46
















0












$begingroup$


Suppose $f(x,y)$ is a function on $mathbb{R}^2$ such that:



1) For a fixed $x$, $f(x,y)$ is a measurable function of $y$ and
2) For a fixed $y$, $f(x,y)$ is a continuous function of $x$.



Prove that for a function $g(y)$ that is measurable on $mathbb{R}$, we have that $f(g(y),y)$ is measurable on $mathbb{R}$.



I can prove that f(x,y)is measurable on $mathbb{R}^2$,but I don't know how to prove this problem?



For f(x,y) is just measurable on $mathbb{R}^2$,is the result still right?










share|cite|improve this question











$endgroup$












  • $begingroup$
    Welcome to MSE. Please read this text about how to ask a good question.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 9:37










  • $begingroup$
    You say that you know how to prove that $f(x,y)$ is measurable. Also $ymapsto (g(y),y)$ is measurable, right? What happens when you take the composition?
    $endgroup$
    – Federico
    Dec 7 '18 at 18:46














0












0








0





$begingroup$


Suppose $f(x,y)$ is a function on $mathbb{R}^2$ such that:



1) For a fixed $x$, $f(x,y)$ is a measurable function of $y$ and
2) For a fixed $y$, $f(x,y)$ is a continuous function of $x$.



Prove that for a function $g(y)$ that is measurable on $mathbb{R}$, we have that $f(g(y),y)$ is measurable on $mathbb{R}$.



I can prove that f(x,y)is measurable on $mathbb{R}^2$,but I don't know how to prove this problem?



For f(x,y) is just measurable on $mathbb{R}^2$,is the result still right?










share|cite|improve this question











$endgroup$




Suppose $f(x,y)$ is a function on $mathbb{R}^2$ such that:



1) For a fixed $x$, $f(x,y)$ is a measurable function of $y$ and
2) For a fixed $y$, $f(x,y)$ is a continuous function of $x$.



Prove that for a function $g(y)$ that is measurable on $mathbb{R}$, we have that $f(g(y),y)$ is measurable on $mathbb{R}$.



I can prove that f(x,y)is measurable on $mathbb{R}^2$,but I don't know how to prove this problem?



For f(x,y) is just measurable on $mathbb{R}^2$,is the result still right?







real-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 7 '18 at 18:35









twnly

867213




867213










asked Dec 7 '18 at 9:33









zhangzhang

11




11












  • $begingroup$
    Welcome to MSE. Please read this text about how to ask a good question.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 9:37










  • $begingroup$
    You say that you know how to prove that $f(x,y)$ is measurable. Also $ymapsto (g(y),y)$ is measurable, right? What happens when you take the composition?
    $endgroup$
    – Federico
    Dec 7 '18 at 18:46


















  • $begingroup$
    Welcome to MSE. Please read this text about how to ask a good question.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 9:37










  • $begingroup$
    You say that you know how to prove that $f(x,y)$ is measurable. Also $ymapsto (g(y),y)$ is measurable, right? What happens when you take the composition?
    $endgroup$
    – Federico
    Dec 7 '18 at 18:46
















$begingroup$
Welcome to MSE. Please read this text about how to ask a good question.
$endgroup$
– José Carlos Santos
Dec 7 '18 at 9:37




$begingroup$
Welcome to MSE. Please read this text about how to ask a good question.
$endgroup$
– José Carlos Santos
Dec 7 '18 at 9:37












$begingroup$
You say that you know how to prove that $f(x,y)$ is measurable. Also $ymapsto (g(y),y)$ is measurable, right? What happens when you take the composition?
$endgroup$
– Federico
Dec 7 '18 at 18:46




$begingroup$
You say that you know how to prove that $f(x,y)$ is measurable. Also $ymapsto (g(y),y)$ is measurable, right? What happens when you take the composition?
$endgroup$
– Federico
Dec 7 '18 at 18:46










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