# Infimum and supremum limits help

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#1

How would I prove the following?

liminf An ∈ F , limsupAn ∈ F , and liminf An ⊆ limsupAn .
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4 years ago
#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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4 years ago
#3
(Original post by Bruce Harrisface)

How would I prove the following?

liminf An ∈ F , limsupAn ∈ F , and liminf An ⊆ limsupAn .
It depends a little bit upon how you have had sigma algebras defined for you. Can you get as far as showing (from the definitions that you have) that a sigma algebra is closed under countable unions and countable intersections?

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 years ago
#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.
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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4 years ago
#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.
Okay; I'm barely able to wrap my head around that but I think I get the general gist of what you're saying, not quite sure how to make the link between liminf, linminf and continuity though, but I'll save that for when I understand more about sigma algebras. Thanks.
0
4 years ago
#6
"Year 9"

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