I was wondering if my approach was correct : The question : Given interval [0,1] and set A = {al

Feinsn 2022-06-25 Answered
I was wondering if my approach was correct :The question :Given interval [0,1] and set A = {all finite and countable subsets of the interval} , what is the σ-algebra generated by A ?
My try:
First , lets look at the 1-element sets - we have ω 1 sets like this with their complements. If we look at the 2-element set and so on we can easily see that we always have the same relation -Cardinality of the set = ω 0 and cardinality of the complement is ω 1 .So my assumption is - The σ-algebra generated from the countable sets is
{ X | | X | = ω 0 | X c | = ω 0 }
But I failed to prove it -
I tried to assume there exist some set generated by the σ-algebra with the property :
| X | = ω 1 | X | = ω 1
Am I correct and I just need to find a way to prove it or I'm wrong ?Thank you in advance.
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Answers (1)

Raven Higgins
Answered 2022-06-26 Author has 17 answers
Not quite: The problem is that your set does not contain complements which are finite.For instance it does not contain the empty set, yet we have: [ 0 , 1 ] c = .So your set is not a sigma-algebra(since sigma-algebras are closed under complements).
So the answer to your question needs to be slightly tweaked, namely the sigma algebra generated by the set of all countable subsets of [0,1], is the subset σ of [0,1] such that elements of σ are exactly those elements of [0,1] which are at most countable or are complements of at most countable sets. It’s easy to check that σ is a sigma-algebra and that it is the smallest sigma-algebra containing the set of all countable subsets of [0,1].

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