Are factors corresponding to a sub-$sigma$-algebra unique?
$begingroup$
Let $X$ be a compact metric space and $mathcal B$ be its Borel $sigma$-algebra.
Let $mu$ be a Borel probability measure on $X$ and $T:Xto X$ me an invertible measure preserving transformation.
Let $U_T:L^2(X, mu)to L^2(X, mu)$ be the associated Koopman operator.
Let $mathcal A$ be the smallest sub-$sigma$-algebra on $X$ with respect to which all the eigenfunctions of $U_T$ are measurable.
Theorem 6.10 in Einsiedler and Ward's Ergodic Theory with a View Towards Number Theory [EW] states the following:
Theorem. The factor of $(X, mathcal B, mu, T)$ corresponding to $mathcal A$ is the largest factor of $(X, mathcal B, mu, T)$ which is isoomorphic to a rotation on some compact abelian group.
I do not follow why are we allowed to use the definite article "the" when asking for a factor corresponding to the sub-$sigma$-algebra $mathcal A$.
I ask this question because the following theorem [EW, Theorem 6.5] only guarantees 'a' factor.
Theorem. Let $(X, mathcal B, mu, T)$ be a measure preserving system, where $X$ is a compact metric space and $mathcal B$ is its Borel $sigma$-algebra.
Let $mathcal A$ be a $T$-invariant sub-$sigma$-algebra on $mathcal B$.
Then there is a measure preserving system $(Y, mathcal B_Y, nu, S)$, where $Y$ is a compact metric space with Borel $sigma$-algebra $mathcal B_Y$, and a factor map $phi:Xto Y$ with $mathcal A=phi^{-1}(mathcal B_Y)pmod mu$.
Is is perhaps true that any two factors corresponding to a given sub-$sigma$-algerba $mathcal A$ are isomorphic?
P.S. I have changed the wording of the theorems I have taken from [EW]. Also, in [EW] the theorems are stated for Borel probability spaces, which are more general than compact metric spaces.
measure-theory ergodic-theory
asked Nov 18 '18 at 17:19
caffeinemachinecaffeinemachine
6,75121458
$endgroup$
|
show 2 more comments
$begingroup$
Let $X$ be a compact metric space and $mathcal B$ be its Borel $sigma$-algebra.
Let $mu$ be a Borel probability measure on $X$ and $T:Xto X$ me an invertible measure preserving transformation.
Let $U_T:L^2(X, mu)to L^2(X, mu)$ be the associated Koopman operator.
Let $mathcal A$ be the smallest sub-$sigma$-algebra on $X$ with respect to which all the eigenfunctions of $U_T$ are measurable.
Theorem 6.10 in Einsiedler and Ward's Ergodic Theory with a View Towards Number Theory [EW] states the following:
Theorem. The factor of $(X, mathcal B, mu, T)$ corresponding to $mathcal A$ is the largest factor of $(X, mathcal B, mu, T)$ which is isoomorphic to a rotation on some compact abelian group.
I do not follow why are we allowed to use the definite article "the" when asking for a factor corresponding to the sub-$sigma$-algebra $mathcal A$.
I ask this question because the following theorem [EW, Theorem 6.5] only guarantees 'a' factor.
Theorem. Let $(X, mathcal B, mu, T)$ be a measure preserving system, where $X$ is a compact metric space and $mathcal B$ is its Borel $sigma$-algebra.
Let $mathcal A$ be a $T$-invariant sub-$sigma$-algebra on $mathcal B$.
Then there is a measure preserving system $(Y, mathcal B_Y, nu, S)$, where $Y$ is a compact metric space with Borel $sigma$-algebra $mathcal B_Y$, and a factor map $phi:Xto Y$ with $mathcal A=phi^{-1}(mathcal B_Y)pmod mu$.
Is is perhaps true that any two factors corresponding to a given sub-$sigma$-algerba $mathcal A$ are isomorphic?
P.S. I have changed the wording of the theorems I have taken from [EW]. Also, in [EW] the theorems are stated for Borel probability spaces, which are more general than compact metric spaces.
measure-theory ergodic-theory
asked Nov 18 '18 at 17:19
caffeinemachinecaffeinemachine
6,75121458
$endgroup$
1
$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:04
$begingroup$
I think your comment solves my issue. Can you frame it as an answer so that I can accept? Thanks.
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:07
$begingroup$
Please read chapter $7$ and answer my question =D
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:10
$begingroup$
@Unpud I don't follow. Your question?
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:25
1
$begingroup$
I was reading the proof of Roth's theorem when I had to go back and read some material from Chapter 6. It will take me some time to start with the proof again since I need to first understand the basics about factors. In case I figure it out I will sure post an answer.
$endgroup$
– caffeinemachine
Nov 19 '18 at 9:03
|
show 2 more comments
$begingroup$
Let $X$ be a compact metric space and $mathcal B$ be its Borel $sigma$-algebra.
Let $mu$ be a Borel probability measure on $X$ and $T:Xto X$ me an invertible measure preserving transformation.
Let $U_T:L^2(X, mu)to L^2(X, mu)$ be the associated Koopman operator.
Let $mathcal A$ be the smallest sub-$sigma$-algebra on $X$ with respect to which all the eigenfunctions of $U_T$ are measurable.
Theorem 6.10 in Einsiedler and Ward's Ergodic Theory with a View Towards Number Theory [EW] states the following:
Theorem. The factor of $(X, mathcal B, mu, T)$ corresponding to $mathcal A$ is the largest factor of $(X, mathcal B, mu, T)$ which is isoomorphic to a rotation on some compact abelian group.
I do not follow why are we allowed to use the definite article "the" when asking for a factor corresponding to the sub-$sigma$-algebra $mathcal A$.
I ask this question because the following theorem [EW, Theorem 6.5] only guarantees 'a' factor.
Theorem. Let $(X, mathcal B, mu, T)$ be a measure preserving system, where $X$ is a compact metric space and $mathcal B$ is its Borel $sigma$-algebra.
Let $mathcal A$ be a $T$-invariant sub-$sigma$-algebra on $mathcal B$.
Then there is a measure preserving system $(Y, mathcal B_Y, nu, S)$, where $Y$ is a compact metric space with Borel $sigma$-algebra $mathcal B_Y$, and a factor map $phi:Xto Y$ with $mathcal A=phi^{-1}(mathcal B_Y)pmod mu$.
Is is perhaps true that any two factors corresponding to a given sub-$sigma$-algerba $mathcal A$ are isomorphic?
P.S. I have changed the wording of the theorems I have taken from [EW]. Also, in [EW] the theorems are stated for Borel probability spaces, which are more general than compact metric spaces.
measure-theory ergodic-theory
asked Nov 18 '18 at 17:19
caffeinemachinecaffeinemachine
6,75121458
$endgroup$
Let $X$ be a compact metric space and $mathcal B$ be its Borel $sigma$-algebra.
Let $mu$ be a Borel probability measure on $X$ and $T:Xto X$ me an invertible measure preserving transformation.
Let $U_T:L^2(X, mu)to L^2(X, mu)$ be the associated Koopman operator.
Let $mathcal A$ be the smallest sub-$sigma$-algebra on $X$ with respect to which all the eigenfunctions of $U_T$ are measurable.
Theorem 6.10 in Einsiedler and Ward's Ergodic Theory with a View Towards Number Theory [EW] states the following:
Theorem. The factor of $(X, mathcal B, mu, T)$ corresponding to $mathcal A$ is the largest factor of $(X, mathcal B, mu, T)$ which is isoomorphic to a rotation on some compact abelian group.
I do not follow why are we allowed to use the definite article "the" when asking for a factor corresponding to the sub-$sigma$-algebra $mathcal A$.
I ask this question because the following theorem [EW, Theorem 6.5] only guarantees 'a' factor.
Theorem. Let $(X, mathcal B, mu, T)$ be a measure preserving system, where $X$ is a compact metric space and $mathcal B$ is its Borel $sigma$-algebra.
Let $mathcal A$ be a $T$-invariant sub-$sigma$-algebra on $mathcal B$.
Then there is a measure preserving system $(Y, mathcal B_Y, nu, S)$, where $Y$ is a compact metric space with Borel $sigma$-algebra $mathcal B_Y$, and a factor map $phi:Xto Y$ with $mathcal A=phi^{-1}(mathcal B_Y)pmod mu$.
Is is perhaps true that any two factors corresponding to a given sub-$sigma$-algerba $mathcal A$ are isomorphic?
P.S. I have changed the wording of the theorems I have taken from [EW]. Also, in [EW] the theorems are stated for Borel probability spaces, which are more general than compact metric spaces.
measure-theory ergodic-theory
measure-theory ergodic-theory
asked Nov 18 '18 at 17:19
caffeinemachinecaffeinemachine
6,75121458
asked Nov 18 '18 at 17:19
caffeinemachinecaffeinemachine
6,75121458
edited Dec 22 '18 at 10:48
caffeinemachine
asked Nov 18 '18 at 17:19
caffeinemachinecaffeinemachine
6,75121458
asked Nov 18 '18 at 17:19
caffeinemachinecaffeinemachine
6,75121458
asked Nov 18 '18 at 17:19
caffeinemachinecaffeinemachine
6,75121458
6,75121458
1
$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:04
$begingroup$
I think your comment solves my issue. Can you frame it as an answer so that I can accept? Thanks.
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:07
$begingroup$
Please read chapter $7$ and answer my question =D
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:10
$begingroup$
@Unpud I don't follow. Your question?
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:25
1
$begingroup$
I was reading the proof of Roth's theorem when I had to go back and read some material from Chapter 6. It will take me some time to start with the proof again since I need to first understand the basics about factors. In case I figure it out I will sure post an answer.
$endgroup$
– caffeinemachine
Nov 19 '18 at 9:03
|
show 2 more comments
1
$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:04
$begingroup$
I think your comment solves my issue. Can you frame it as an answer so that I can accept? Thanks.
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:07
$begingroup$
Please read chapter $7$ and answer my question =D
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:10
$begingroup$
@Unpud I don't follow. Your question?
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:25
1
$begingroup$
I was reading the proof of Roth's theorem when I had to go back and read some material from Chapter 6. It will take me some time to start with the proof again since I need to first understand the basics about factors. In case I figure it out I will sure post an answer.
$endgroup$
– caffeinemachine
Nov 19 '18 at 9:03
1
1
$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:04
$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:04
$begingroup$
I think your comment solves my issue. Can you frame it as an answer so that I can accept? Thanks.
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:07
$begingroup$
I think your comment solves my issue. Can you frame it as an answer so that I can accept? Thanks.
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:07
$begingroup$
Please read chapter $7$ and answer my question =D
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:10
$begingroup$
Please read chapter $7$ and answer my question =D
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:10
$begingroup$
@Unpud I don't follow. Your question?
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:25
$begingroup$
@Unpud I don't follow. Your question?
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:25
1
1
$begingroup$
I was reading the proof of Roth's theorem when I had to go back and read some material from Chapter 6. It will take me some time to start with the proof again since I need to first understand the basics about factors. In case I figure it out I will sure post an answer.
$endgroup$
– caffeinemachine
Nov 19 '18 at 9:03
$begingroup$
I was reading the proof of Roth's theorem when I had to go back and read some material from Chapter 6. It will take me some time to start with the proof again since I need to first understand the basics about factors. In case I figure it out I will sure post an answer.
$endgroup$
– caffeinemachine
Nov 19 '18 at 9:03
|
show 2 more comments
1 Answer
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$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
answered Nov 18 '18 at 19:09
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
404420
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$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
answered Nov 18 '18 at 19:09
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
404420
$endgroup$
add a comment |
$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
answered Nov 18 '18 at 19:09
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
404420
$endgroup$
add a comment |
$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
answered Nov 18 '18 at 19:09
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
404420
$endgroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
answered Nov 18 '18 at 19:09
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
404420
answered Nov 18 '18 at 19:09
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
404420
answered Nov 18 '18 at 19:09
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
404420
answered Nov 18 '18 at 19:09
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
404420
404420
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$begingroup$
Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:04
$begingroup$
I think your comment solves my issue. Can you frame it as an answer so that I can accept? Thanks.
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:07
$begingroup$
Please read chapter $7$ and answer my question =D
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 18 '18 at 19:10
$begingroup$
@Unpud I don't follow. Your question?
$endgroup$
– caffeinemachine
Nov 18 '18 at 19:25
1
$begingroup$
I was reading the proof of Roth's theorem when I had to go back and read some material from Chapter 6. It will take me some time to start with the proof again since I need to first understand the basics about factors. In case I figure it out I will sure post an answer.
$endgroup$
– caffeinemachine
Nov 19 '18 at 9:03