Can a $\sigma$-algebra in a set $X$ have cardinality $\aleph_0$, the cardinality of the naturals?
I do not have a clue on how to start with this? Can someone please give me a hint?
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1While the title of the question is about uncountable underlying sets; the answers do not make this separation. – Asaf Karagila Apr 05 '15 at 22:23
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@AsafKaragila: Thanks for the link. I was not aware of this – Silver moon Apr 05 '15 at 22:27
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Pick a measurable. Then either it can be partitioned into two measurables or its complement can. If one of them doesn't can imagine looking at the one that can. Repeat inductively.
This gives you a sequence of disjoint measurables. But now any subsequence of them can be joined and get a different measurable. The number of subsequences is uncountable.
Karanko
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