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.

Question

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&#039;m wrong ?Thank you in advance.

Raven Higgins · Accepted Answer

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].

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

Answered question

Answer & Explanation

New Questions in Pre-Algebra