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?
real-analysis
edited Dec 7 '18 at 18:35
twnly
857213
asked Dec 7 '18 at 9:33
zhangzhang
11
$endgroup$
add a comment |
$begingroup$
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?
real-analysis
edited Dec 7 '18 at 18:35
twnly
857213
asked Dec 7 '18 at 9:33
zhangzhang
11
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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?
real-analysis
edited Dec 7 '18 at 18:35
twnly
857213
asked Dec 7 '18 at 9:33
zhangzhang
11
$endgroup$
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
real-analysis
edited Dec 7 '18 at 18:35
twnly
857213
asked Dec 7 '18 at 9:33
zhangzhang
11
edited Dec 7 '18 at 18:35
twnly
857213
asked Dec 7 '18 at 9:33
zhangzhang
11
edited Dec 7 '18 at 18:35
twnly
857213
edited Dec 7 '18 at 18:35
twnly
857213
edited Dec 7 '18 at 18:35
twnly
857213
857213
asked Dec 7 '18 at 9:33
zhangzhang
11
asked Dec 7 '18 at 9:33
zhangzhang
11
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
add a comment |
$begingroup$
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
add a comment |
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$begingroup$
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