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How would I prove the following?
liminf An ∈ F , limsupAn ∈ F , and liminf An ⊆ limsupAn .
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#2
Does the Borel-Cantelli or its counterpart help in anyway? Probably not, I barely understand what your question is even asking to be honest.
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#3
(Original post by Bruce Harrisface)
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How would I prove the following?
liminf An ∈ F , limsupAn ∈ F , and liminf An ⊆ limsupAn .
How would I prove the following?
liminf An ∈ F , limsupAn ∈ F , and liminf An ⊆ limsupAn .
If so, you should be able to write down the definition of limsup and liminf and show that one is a countable union of countable intersections and the other is a countable intersection of countable unions.
To show the final part, can you see that limsup An is the set of points that are in infinitely many of the An and that liminf An is the set of points that fail to be in at most a finite number of the An?
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#4
(Original post by Zacken)
Does the Borel-Cantelli or its counterpart help in anyway? Probably not, I barely understand what your question is even asking to be honest.
Does the Borel-Cantelli or its counterpart help in anyway? Probably not, I barely understand what your question is even asking to be honest.
The idea of a sigma algebra is that it's the "natural" place to define a measure that is going to be countably additive - therefore you need it to be closed under countable unions and intersections; liminf and limsup come in useful once you've got the measure defined and you're interested in its "continuity" properties.
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#5
(Original post by Gregorius)
Borel-Cantelli is about the behaviour of a measure defined on a sigma algebra - so it's a stage beyond what this question is asking for.
The idea of a sigma algebra is that it's the "natural" place to define a measure that is going to be countably additive - therefore you need it to be closed under countable unions and intersections; liminf and limsup come in useful once you've got the measure defined and you're interested in its "continuity" properties.
Borel-Cantelli is about the behaviour of a measure defined on a sigma algebra - so it's a stage beyond what this question is asking for.
The idea of a sigma algebra is that it's the "natural" place to define a measure that is going to be countably additive - therefore you need it to be closed under countable unions and intersections; liminf and limsup come in useful once you've got the measure defined and you're interested in its "continuity" properties.
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