Can any sense be made of the following conjecture? For any given $x$ in $[0,1]$, “most” measure 1 subsets...
2
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Can any sense be made of the following conjecture? For any given $x$ in $[0,1]$, "most" measure $1$ subsets of $[0,1]$ contain $x$. More generally what kinds of non-trivial measures exist on sets of sets of reals?
measure-theory
edited Dec 17 '18 at 1:57
Lau
517315
asked Dec 17 '18 at 0:36
Stephen YabloStephen Yablo
192
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closed as off-topic by Eevee Trainer, kjetil b halvorsen, Did, Saad, Tianlalu Dec 17 '18 at 3:42
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." – Eevee Trainer, kjetil b halvorsen, Did, Saad, Tianlalu
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
2
$begingroup$
Can any sense be made of the following conjecture? For any given $x$ in $[0,1]$, "most" measure $1$ subsets of $[0,1]$ contain $x$. More generally what kinds of non-trivial measures exist on sets of sets of reals?
measure-theory
edited Dec 17 '18 at 1:57
Lau
517315
asked Dec 17 '18 at 0:36
Stephen YabloStephen Yablo
192
$endgroup$
closed as off-topic by Eevee Trainer, kjetil b halvorsen, Did, Saad, Tianlalu Dec 17 '18 at 3:42
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." – Eevee Trainer, kjetil b halvorsen, Did, Saad, Tianlalu
If this question can be reworded to fit the rules in the help center, please edit the question.
2
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Ridiculous that this is being voted to close. What context do people need here? This isn't an elementary homework problem.
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– MathematicsStudent1122
Dec 17 '18 at 2:07
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Point process are an example, such as poisson process, or spectrum of a random (psd) matrix. The ones I've seen are almost surely locally finite, but you can easily make up some that aren't (like, fill in every other interval created by the points of the process). I guess you want some kind of natural measure. One property that I think it should have is that if you condition the subset on lying in a subinterval, the process should look like the original process (after an affine rescaling). (This is to be like the uniform measure.)
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– Lorenzo
Dec 17 '18 at 2:34
$begingroup$
The process that picks a uniform point x and declares that the set has that restriction property (i.e. conditioning on lying in a subinterval and then rescaling doesn't change the measure), so probably you want to also require that the random subset is almost surely uncountable. I don't know if any measures satisfying both those properties exist. (P.S. The inspiration for that 'restriction property' as a way to emulate uniformity is this paper: arxiv.org/pdf/math/0204277.pdf )
$endgroup$
– Lorenzo
Dec 17 '18 at 2:39
add a comment |
2
2
2
$begingroup$
Can any sense be made of the following conjecture? For any given $x$ in $[0,1]$, "most" measure $1$ subsets of $[0,1]$ contain $x$. More generally what kinds of non-trivial measures exist on sets of sets of reals?
measure-theory
edited Dec 17 '18 at 1:57
Lau
517315
asked Dec 17 '18 at 0:36
Stephen YabloStephen Yablo
192
$endgroup$
Can any sense be made of the following conjecture? For any given $x$ in $[0,1]$, "most" measure $1$ subsets of $[0,1]$ contain $x$. More generally what kinds of non-trivial measures exist on sets of sets of reals?
measure-theory
measure-theory
edited Dec 17 '18 at 1:57
Lau
517315
asked Dec 17 '18 at 0:36
Stephen YabloStephen Yablo
192
edited Dec 17 '18 at 1:57
Lau
517315
asked Dec 17 '18 at 0:36
Stephen YabloStephen Yablo
192
edited Dec 17 '18 at 1:57
Lau
517315
edited Dec 17 '18 at 1:57
Lau
517315
edited Dec 17 '18 at 1:57
Lau
517315
517315
asked Dec 17 '18 at 0:36
Stephen YabloStephen Yablo
192
asked Dec 17 '18 at 0:36
Stephen YabloStephen Yablo
192
asked Dec 17 '18 at 0:36
Stephen YabloStephen Yablo
192
192
closed as off-topic by Eevee Trainer, kjetil b halvorsen, Did, Saad, Tianlalu Dec 17 '18 at 3:42
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." – Eevee Trainer, kjetil b halvorsen, Did, Saad, Tianlalu
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Eevee Trainer, kjetil b halvorsen, Did, Saad, Tianlalu Dec 17 '18 at 3:42
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." – Eevee Trainer, kjetil b halvorsen, Did, Saad, Tianlalu
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Ridiculous that this is being voted to close. What context do people need here? This isn't an elementary homework problem.
$endgroup$
– MathematicsStudent1122
Dec 17 '18 at 2:07
$begingroup$
Point process are an example, such as poisson process, or spectrum of a random (psd) matrix. The ones I've seen are almost surely locally finite, but you can easily make up some that aren't (like, fill in every other interval created by the points of the process). I guess you want some kind of natural measure. One property that I think it should have is that if you condition the subset on lying in a subinterval, the process should look like the original process (after an affine rescaling). (This is to be like the uniform measure.)
$endgroup$
– Lorenzo
Dec 17 '18 at 2:34
$begingroup$
The process that picks a uniform point x and declares that the set has that restriction property (i.e. conditioning on lying in a subinterval and then rescaling doesn't change the measure), so probably you want to also require that the random subset is almost surely uncountable. I don't know if any measures satisfying both those properties exist. (P.S. The inspiration for that 'restriction property' as a way to emulate uniformity is this paper: arxiv.org/pdf/math/0204277.pdf )
$endgroup$
– Lorenzo
Dec 17 '18 at 2:39
add a comment |
2
$begingroup$
Ridiculous that this is being voted to close. What context do people need here? This isn't an elementary homework problem.
$endgroup$
– MathematicsStudent1122
Dec 17 '18 at 2:07
$begingroup$
Point process are an example, such as poisson process, or spectrum of a random (psd) matrix. The ones I've seen are almost surely locally finite, but you can easily make up some that aren't (like, fill in every other interval created by the points of the process). I guess you want some kind of natural measure. One property that I think it should have is that if you condition the subset on lying in a subinterval, the process should look like the original process (after an affine rescaling). (This is to be like the uniform measure.)
$endgroup$
– Lorenzo
Dec 17 '18 at 2:34
$begingroup$
The process that picks a uniform point x and declares that the set has that restriction property (i.e. conditioning on lying in a subinterval and then rescaling doesn't change the measure), so probably you want to also require that the random subset is almost surely uncountable. I don't know if any measures satisfying both those properties exist. (P.S. The inspiration for that 'restriction property' as a way to emulate uniformity is this paper: arxiv.org/pdf/math/0204277.pdf )
$endgroup$
– Lorenzo
Dec 17 '18 at 2:39
2
2
$begingroup$
Ridiculous that this is being voted to close. What context do people need here? This isn't an elementary homework problem.
$endgroup$
– MathematicsStudent1122
Dec 17 '18 at 2:07
$begingroup$
Ridiculous that this is being voted to close. What context do people need here? This isn't an elementary homework problem.
$endgroup$
– MathematicsStudent1122
Dec 17 '18 at 2:07
$begingroup$
Point process are an example, such as poisson process, or spectrum of a random (psd) matrix. The ones I've seen are almost surely locally finite, but you can easily make up some that aren't (like, fill in every other interval created by the points of the process). I guess you want some kind of natural measure. One property that I think it should have is that if you condition the subset on lying in a subinterval, the process should look like the original process (after an affine rescaling). (This is to be like the uniform measure.)
$endgroup$
– Lorenzo
Dec 17 '18 at 2:34
$begingroup$
Point process are an example, such as poisson process, or spectrum of a random (psd) matrix. The ones I've seen are almost surely locally finite, but you can easily make up some that aren't (like, fill in every other interval created by the points of the process). I guess you want some kind of natural measure. One property that I think it should have is that if you condition the subset on lying in a subinterval, the process should look like the original process (after an affine rescaling). (This is to be like the uniform measure.)
$endgroup$
– Lorenzo
Dec 17 '18 at 2:34
$begingroup$
The process that picks a uniform point x and declares that the set has that restriction property (i.e. conditioning on lying in a subinterval and then rescaling doesn't change the measure), so probably you want to also require that the random subset is almost surely uncountable. I don't know if any measures satisfying both those properties exist. (P.S. The inspiration for that 'restriction property' as a way to emulate uniformity is this paper: arxiv.org/pdf/math/0204277.pdf )
$endgroup$
– Lorenzo
Dec 17 '18 at 2:39
$begingroup$
The process that picks a uniform point x and declares that the set has that restriction property (i.e. conditioning on lying in a subinterval and then rescaling doesn't change the measure), so probably you want to also require that the random subset is almost surely uncountable. I don't know if any measures satisfying both those properties exist. (P.S. The inspiration for that 'restriction property' as a way to emulate uniformity is this paper: arxiv.org/pdf/math/0204277.pdf )
$endgroup$
– Lorenzo
Dec 17 '18 at 2:39
add a comment |
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Ridiculous that this is being voted to close. What context do people need here? This isn't an elementary homework problem.
$endgroup$
– MathematicsStudent1122
Dec 17 '18 at 2:07
$begingroup$
Point process are an example, such as poisson process, or spectrum of a random (psd) matrix. The ones I've seen are almost surely locally finite, but you can easily make up some that aren't (like, fill in every other interval created by the points of the process). I guess you want some kind of natural measure. One property that I think it should have is that if you condition the subset on lying in a subinterval, the process should look like the original process (after an affine rescaling). (This is to be like the uniform measure.)
$endgroup$
– Lorenzo
Dec 17 '18 at 2:34
$begingroup$
The process that picks a uniform point x and declares that the set has that restriction property (i.e. conditioning on lying in a subinterval and then rescaling doesn't change the measure), so probably you want to also require that the random subset is almost surely uncountable. I don't know if any measures satisfying both those properties exist. (P.S. The inspiration for that 'restriction property' as a way to emulate uniformity is this paper: arxiv.org/pdf/math/0204277.pdf )
$endgroup$
– Lorenzo
Dec 17 '18 at 2:39