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?
2026-04-03 01:38:41.1775180321
$\sigma$-algebra with cardinality $\aleph_0$
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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.