Two examples show that $L^2$ convergence and almost everywhere convergence are independent notions [closed]
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To show the main difference between pointwise ergodic theorem and mean ergodic theorem we have to show that $L^2$ convergence and almost everywhere convergence are independent notions.
Therefore, I am looking for two examples of sequences of functions, the first sequence is $L^2$ convergent but not almost everywhere convergent and the second one is almost everywhere convergent but not $L^2$convergent.
functional-analysis measure-theory convergence ergodic-theory
asked Jan 8 at 14:41
Neil hawkingNeil hawking
48819
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closed as off-topic by Xander Henderson, Holo, amWhy, Did, Davide Giraudo Jan 8 at 15:45
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, Did, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
To show the main difference between pointwise ergodic theorem and mean ergodic theorem we have to show that $L^2$ convergence and almost everywhere convergence are independent notions.
Therefore, I am looking for two examples of sequences of functions, the first sequence is $L^2$ convergent but not almost everywhere convergent and the second one is almost everywhere convergent but not $L^2$convergent.
functional-analysis measure-theory convergence ergodic-theory
asked Jan 8 at 14:41
Neil hawkingNeil hawking
48819
$endgroup$
closed as off-topic by Xander Henderson, Holo, amWhy, Did, Davide Giraudo Jan 8 at 15:45
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, Did, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
3
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You have asked two questions here. The first is "Does convergence in $L^2$ imply almost everywhere convergence?" The second is "Does almost everywhere convergence imply $L^2$ convergence?" Both questions have been answered before.
$endgroup$
– Xander Henderson
Jan 8 at 14:49
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And zero personal input.
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– Did
Jan 8 at 15:06
add a comment |
$begingroup$
To show the main difference between pointwise ergodic theorem and mean ergodic theorem we have to show that $L^2$ convergence and almost everywhere convergence are independent notions.
Therefore, I am looking for two examples of sequences of functions, the first sequence is $L^2$ convergent but not almost everywhere convergent and the second one is almost everywhere convergent but not $L^2$convergent.
functional-analysis measure-theory convergence ergodic-theory
asked Jan 8 at 14:41
Neil hawkingNeil hawking
48819
$endgroup$
To show the main difference between pointwise ergodic theorem and mean ergodic theorem we have to show that $L^2$ convergence and almost everywhere convergence are independent notions.
Therefore, I am looking for two examples of sequences of functions, the first sequence is $L^2$ convergent but not almost everywhere convergent and the second one is almost everywhere convergent but not $L^2$convergent.
functional-analysis measure-theory convergence ergodic-theory
functional-analysis measure-theory convergence ergodic-theory
asked Jan 8 at 14:41
Neil hawkingNeil hawking
48819
asked Jan 8 at 14:41
Neil hawkingNeil hawking
48819
asked Jan 8 at 14:41
Neil hawkingNeil hawking
48819
asked Jan 8 at 14:41
Neil hawkingNeil hawking
48819
asked Jan 8 at 14:41
Neil hawkingNeil hawking
48819
48819
closed as off-topic by Xander Henderson, Holo, amWhy, Did, Davide Giraudo Jan 8 at 15:45
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, Did, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Xander Henderson, Holo, amWhy, Did, Davide Giraudo Jan 8 at 15:45
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, Did, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
3
$begingroup$
You have asked two questions here. The first is "Does convergence in $L^2$ imply almost everywhere convergence?" The second is "Does almost everywhere convergence imply $L^2$ convergence?" Both questions have been answered before.
$endgroup$
– Xander Henderson
Jan 8 at 14:49
$begingroup$
And zero personal input.
$endgroup$
– Did
Jan 8 at 15:06
add a comment |
3
$begingroup$
You have asked two questions here. The first is "Does convergence in $L^2$ imply almost everywhere convergence?" The second is "Does almost everywhere convergence imply $L^2$ convergence?" Both questions have been answered before.
$endgroup$
– Xander Henderson
Jan 8 at 14:49
$begingroup$
And zero personal input.
$endgroup$
– Did
Jan 8 at 15:06
3
3
$begingroup$
You have asked two questions here. The first is "Does convergence in $L^2$ imply almost everywhere convergence?" The second is "Does almost everywhere convergence imply $L^2$ convergence?" Both questions have been answered before.
$endgroup$
– Xander Henderson
Jan 8 at 14:49
$begingroup$
You have asked two questions here. The first is "Does convergence in $L^2$ imply almost everywhere convergence?" The second is "Does almost everywhere convergence imply $L^2$ convergence?" Both questions have been answered before.
$endgroup$
– Xander Henderson
Jan 8 at 14:49
$begingroup$
And zero personal input.
$endgroup$
– Did
Jan 8 at 15:06
$begingroup$
And zero personal input.
$endgroup$
– Did
Jan 8 at 15:06
add a comment |
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$begingroup$
You have asked two questions here. The first is "Does convergence in $L^2$ imply almost everywhere convergence?" The second is "Does almost everywhere convergence imply $L^2$ convergence?" Both questions have been answered before.
$endgroup$
– Xander Henderson
Jan 8 at 14:49
$begingroup$
And zero personal input.
$endgroup$
– Did
Jan 8 at 15:06