Mixing
Prove that composition of functions is measurable
$begingroup$
Let $f: X to [0, infty)$ and $p:X to mathbb{R}$ be measurable functions. Prove that $g(x) = [f(x)]^{p(x)}$ is a measurable function.
My plan was invoking the following theorem:
For $(X,M)$ a measurable space, $(Y, tau_Y), (Z, tau_Z)$ topological spaces, if $f: X to Y$ is measurable and $g: Y to Z$ is continuous, then $h = g circ f$ is measurable.
Now, I defined the function
$$phi: [0,infty)times mathbb{R} to [0, infty)$$
$$phi(f(x),p(x)) := [f(x)]^{p(x)}$$
Now I just have to prove continuity of $phi$, but I am having trouble doing this.
real-analysis general-topology measure-theory
asked Feb 1 at 8:15
The BoscoThe Bosco
608212
$endgroup$
add a comment |
$begingroup$
Let $f: X to [0, infty)$ and $p:X to mathbb{R}$ be measurable functions. Prove that $g(x) = [f(x)]^{p(x)}$ is a measurable function.
My plan was invoking the following theorem:
For $(X,M)$ a measurable space, $(Y, tau_Y), (Z, tau_Z)$ topological spaces, if $f: X to Y$ is measurable and $g: Y to Z$ is continuous, then $h = g circ f$ is measurable.
Now, I defined the function
$$phi: [0,infty)times mathbb{R} to [0, infty)$$
$$phi(f(x),p(x)) := [f(x)]^{p(x)}$$
Now I just have to prove continuity of $phi$, but I am having trouble doing this.
real-analysis general-topology measure-theory
asked Feb 1 at 8:15
The BoscoThe Bosco
608212
$endgroup$
add a comment |
$begingroup$
Let $f: X to [0, infty)$ and $p:X to mathbb{R}$ be measurable functions. Prove that $g(x) = [f(x)]^{p(x)}$ is a measurable function.
My plan was invoking the following theorem:
For $(X,M)$ a measurable space, $(Y, tau_Y), (Z, tau_Z)$ topological spaces, if $f: X to Y$ is measurable and $g: Y to Z$ is continuous, then $h = g circ f$ is measurable.
Now, I defined the function
$$phi: [0,infty)times mathbb{R} to [0, infty)$$
$$phi(f(x),p(x)) := [f(x)]^{p(x)}$$
Now I just have to prove continuity of $phi$, but I am having trouble doing this.
real-analysis general-topology measure-theory
asked Feb 1 at 8:15
The BoscoThe Bosco
608212
$endgroup$
Let $f: X to [0, infty)$ and $p:X to mathbb{R}$ be measurable functions. Prove that $g(x) = [f(x)]^{p(x)}$ is a measurable function.
My plan was invoking the following theorem:
For $(X,M)$ a measurable space, $(Y, tau_Y), (Z, tau_Z)$ topological spaces, if $f: X to Y$ is measurable and $g: Y to Z$ is continuous, then $h = g circ f$ is measurable.
Now, I defined the function
$$phi: [0,infty)times mathbb{R} to [0, infty)$$
$$phi(f(x),p(x)) := [f(x)]^{p(x)}$$
Now I just have to prove continuity of $phi$, but I am having trouble doing this.
real-analysis general-topology measure-theory
real-analysis general-topology measure-theory
asked Feb 1 at 8:15
The BoscoThe Bosco
608212
asked Feb 1 at 8:15
The BoscoThe Bosco
608212
asked Feb 1 at 8:15
The BoscoThe Bosco
608212
asked Feb 1 at 8:15
The BoscoThe Bosco
608212
asked Feb 1 at 8:15
The BoscoThe Bosco
608212
608212
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
Your function $phi$ is not continuous!. In fact, $(frac 1 n, -1) to (0,-1)$ but $phi (frac 1 n, -1) to infty$. Instead of this approach write $g(x)$ as $e^{p(x)log (f(x))}$ and note that $log (f(x))$ is an extended real valued measurable function. Can you complete the argument now?
answered Feb 1 at 8:26
Kavi Rama MurthyKavi Rama Murthy
73.8k53170
$endgroup$
$begingroup$
What is it with sequences of the form $frac{1}{n}$ (or very similar) that they keep popping up all over the place in measure theory?
$endgroup$
– The Bosco
Feb 1 at 8:48
$begingroup$
They pop up everywhere in Mathematics, not just measure Theory!
$endgroup$
– Kavi Rama Murthy
Feb 1 at 8:52
$begingroup$
But I've seen them way more often in this branch! Also, what you meant was that I should substitute my $phi$ function with $g$ or the $g$ in the theorem I wrote?
$endgroup$
– The Bosco
Feb 1 at 9:11
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your function $phi$ is not continuous!. In fact, $(frac 1 n, -1) to (0,-1)$ but $phi (frac 1 n, -1) to infty$. Instead of this approach write $g(x)$ as $e^{p(x)log (f(x))}$ and note that $log (f(x))$ is an extended real valued measurable function. Can you complete the argument now?
answered Feb 1 at 8:26
Kavi Rama MurthyKavi Rama Murthy
73.8k53170
$endgroup$
$begingroup$
What is it with sequences of the form $frac{1}{n}$ (or very similar) that they keep popping up all over the place in measure theory?
$endgroup$
– The Bosco
Feb 1 at 8:48
$begingroup$
They pop up everywhere in Mathematics, not just measure Theory!
$endgroup$
– Kavi Rama Murthy
Feb 1 at 8:52
$begingroup$
But I've seen them way more often in this branch! Also, what you meant was that I should substitute my $phi$ function with $g$ or the $g$ in the theorem I wrote?
$endgroup$
– The Bosco
Feb 1 at 9:11
add a comment |
$begingroup$
Your function $phi$ is not continuous!. In fact, $(frac 1 n, -1) to (0,-1)$ but $phi (frac 1 n, -1) to infty$. Instead of this approach write $g(x)$ as $e^{p(x)log (f(x))}$ and note that $log (f(x))$ is an extended real valued measurable function. Can you complete the argument now?
answered Feb 1 at 8:26
Kavi Rama MurthyKavi Rama Murthy
73.8k53170
$endgroup$
$begingroup$
What is it with sequences of the form $frac{1}{n}$ (or very similar) that they keep popping up all over the place in measure theory?
$endgroup$
– The Bosco
Feb 1 at 8:48
$begingroup$
They pop up everywhere in Mathematics, not just measure Theory!
$endgroup$
– Kavi Rama Murthy
Feb 1 at 8:52
$begingroup$
But I've seen them way more often in this branch! Also, what you meant was that I should substitute my $phi$ function with $g$ or the $g$ in the theorem I wrote?
$endgroup$
– The Bosco
Feb 1 at 9:11
add a comment |
$begingroup$
Your function $phi$ is not continuous!. In fact, $(frac 1 n, -1) to (0,-1)$ but $phi (frac 1 n, -1) to infty$. Instead of this approach write $g(x)$ as $e^{p(x)log (f(x))}$ and note that $log (f(x))$ is an extended real valued measurable function. Can you complete the argument now?
answered Feb 1 at 8:26
Kavi Rama MurthyKavi Rama Murthy
73.8k53170
$endgroup$
Your function $phi$ is not continuous!. In fact, $(frac 1 n, -1) to (0,-1)$ but $phi (frac 1 n, -1) to infty$. Instead of this approach write $g(x)$ as $e^{p(x)log (f(x))}$ and note that $log (f(x))$ is an extended real valued measurable function. Can you complete the argument now?
answered Feb 1 at 8:26
Kavi Rama MurthyKavi Rama Murthy
73.8k53170
answered Feb 1 at 8:26
Kavi Rama MurthyKavi Rama Murthy
73.8k53170
answered Feb 1 at 8:26
Kavi Rama MurthyKavi Rama Murthy
73.8k53170
answered Feb 1 at 8:26
Kavi Rama MurthyKavi Rama Murthy
73.8k53170
73.8k53170
$begingroup$
What is it with sequences of the form $frac{1}{n}$ (or very similar) that they keep popping up all over the place in measure theory?
$endgroup$
– The Bosco
Feb 1 at 8:48
$begingroup$
They pop up everywhere in Mathematics, not just measure Theory!
$endgroup$
– Kavi Rama Murthy
Feb 1 at 8:52
$begingroup$
But I've seen them way more often in this branch! Also, what you meant was that I should substitute my $phi$ function with $g$ or the $g$ in the theorem I wrote?
$endgroup$
– The Bosco
Feb 1 at 9:11
add a comment |
$begingroup$
What is it with sequences of the form $frac{1}{n}$ (or very similar) that they keep popping up all over the place in measure theory?
$endgroup$
– The Bosco
Feb 1 at 8:48
$begingroup$
They pop up everywhere in Mathematics, not just measure Theory!
$endgroup$
– Kavi Rama Murthy
Feb 1 at 8:52
$begingroup$
But I've seen them way more often in this branch! Also, what you meant was that I should substitute my $phi$ function with $g$ or the $g$ in the theorem I wrote?
$endgroup$
– The Bosco
Feb 1 at 9:11
$begingroup$
What is it with sequences of the form $frac{1}{n}$ (or very similar) that they keep popping up all over the place in measure theory?
$endgroup$
– The Bosco
Feb 1 at 8:48
$begingroup$
What is it with sequences of the form $frac{1}{n}$ (or very similar) that they keep popping up all over the place in measure theory?
$endgroup$
– The Bosco
Feb 1 at 8:48
$begingroup$
They pop up everywhere in Mathematics, not just measure Theory!
$endgroup$
– Kavi Rama Murthy
Feb 1 at 8:52
$begingroup$
They pop up everywhere in Mathematics, not just measure Theory!
$endgroup$
– Kavi Rama Murthy
Feb 1 at 8:52
$begingroup$
But I've seen them way more often in this branch! Also, what you meant was that I should substitute my $phi$ function with $g$ or the $g$ in the theorem I wrote?
$endgroup$
– The Bosco
Feb 1 at 9:11
$begingroup$
But I've seen them way more often in this branch! Also, what you meant was that I should substitute my $phi$ function with $g$ or the $g$ in the theorem I wrote?
$endgroup$
– The Bosco
Feb 1 at 9:11
add a comment |
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