Infinite sigma-algebra

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I'm supposed to answer the question "Can a sigma-algebra be infinite and countable?"
I think I can show that if it has a countable number of disjoint subsets, then it can't be countable considering the possible combinations of the subsets.
Now I need to show that if a sigma-algebra consists of an infinite number of subsets, then it has a countable number of disjoint subsets.
Any ideas on how I can do this?
 

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  • #3
Hurkyl
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I think I can show that if it has a countable number of disjoint subsets, then it can't be countable considering the possible combinations of the subsets.
No you can't.

Now, if you instead said countably infinite... :smile:


Now I need to show that if a sigma-algebra consists of an infinite number of subsets, then it has a [countably infinite] number of disjoint subsets.
(I edited it)

Proof by contradiction, maybe?
 
  • #4
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Hurkyl said:
No you can't.

Now, if you instead said countably infinite... :smile:
That's what I meant. I was sloppy. :redface:
Thank you both for the help.
 

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