Is there an infinite countable $\sigma$-algebra on an uncountable set

Let $\Omega$ be a set.

If $\Omega$ is finite, then any $\sigma$-algebra on $\Omega$ is finite.

If $\Omega$ is infinite and countable, a $\sigma$-algebra on $\Omega$ cannot be infinite and countable.

What about if $\Omega$ is not countable ? Is it possible to find an uncountable $\Omega$ with a $\sigma$-algebra that is infinite and countable ?

Solution 1:

Suppose that $\lvert \Omega\rvert\ge\aleph_0$, and $\mathscr M\subset\mathscr P(\Omega)$ is a $\sigma-$algebra. We shall show that: $$ \textit{Either}\,\,\,\, \lvert\mathscr M\rvert<\aleph_0\quad or\quad \lvert\mathscr M\rvert\ge 2^{\aleph_0}. $$ Define in $\Omega$ the following relation: $$ a\sim b\qquad\text{iff}\qquad \forall E\in\mathscr M\, (\,a\in E\Longleftrightarrow b\in E\,). $$ Clearly, "$\sim$" is an equivalence relation in $\Omega$, and every $E\in\mathscr M$ is a union of equivalence classes. Also, for every two different classes $[a]$ and $[b]$, there are $E,F\in\mathscr M$, with $E\cap F=\varnothing$, such that $[a]\subset E$ and $[b]\subset F$.

Case I. If there are finitely many classes, say $n$, then each class belongs to $\mathscr M$, and clearly $\lvert \mathscr M\rvert=2^n$.

Case II. Assume there are $\aleph_0$ classes. Fix a class $[a]$, and let $\{[a_n]:n\in\mathbb N\}$ be the remaining classes. For every $n\in\mathbb N$, there exist $E_n,F_n\mathscr\in M$, such that $[a]\subset E_n$, $[a_n]\subset F_n$ and $E_n\cap F_n=\varnothing$. Clearly, $[a]=\bigcap_{n\in\mathbb N} E_n\in\mathscr M$, and thus $\lvert \mathscr M\rvert=2^{\aleph_0}$.

Case III. If there are uncountably many classes, we can pick infinite countable of them $[a_n]$, $n\in\mathbb N$, and disjoint sets $E_n\in\mathscr M$, with $[a_n]\subset E_n$, (using the Axiom of Choice), and then realise that the $\sigma-$algebra generated by the $E_n$'s has the cardinality of the continuum and is a subalgebra of $\mathscr M$.

Solution 2:

No. The same proof is used as in the case where $\Omega$ is countable and uncountable.

If $\cal B$ is an infinite $\sigma$-algebra, then $\cal B$ has at least the cardinality of the continuum.

The proof is as follows, since $\cal B$ is infinite, it has a countable subset $\{A_i\mid i\in\Bbb N\}$. If this countable set is a $\subseteq$-chain (without loss of generality, it's increasing) take $B_i=A_i\setminus\bigcup_{j<i}A_j$, and this is an infinite family of pairwise disjoint sets. Otherwise, it's not a chain, and without loss of generality doesn't contain an infinite chain either, and by similar a induction (although now it's gonna be slightly dirtier, you might have to skip a few indices, between $B_i$ and $B_{i+1}$) create an infinite family $B_i$ of pairwise disjoint non-empty sets.

Now it's easy to show that $\cal B$ has at least the cardinality of the continuum. If $D\subseteq\Bbb N$, take $B_D=\bigcup_{i\in D}B_i$. And it's quite easy to see that $D\neq D'$ implies that $B_D\neq B_{D'}$. $\quad \square$

A word on choice. $\tiny\textsf{(with regards to tomasz)}$

Note that this proof uses the axiom of choice. We used the fact that if $\cal B$ is infinite, then it has a countably infinite subset. In fact it is consistent that there is a $\sigma$-algebra which is infinite, but has no countably infinite subset. Of course this $\sigma$-algebra is not countable either.

If we are only interested in the answer to the original question, then the axiom of choice is not used anywhere. If there is a countably infinite subset, then the proof follows (note that all the choices above, except the $A_i$'s, are done by induction on the chosen sequence, so they are in fact AC-free); and if there is no countably infinite subset, then certainly $\cal B$ is not countable!

(One example of such $\sigma$-algebra that has no countably infinite subset, is the power set of an amorphous set; where amorphous means that every subset is finite or its complement is finite. Why is this a $\sigma$-algebra? From a countably infinite collection of subsets we can define an infinite co-infinite subset, in a way similar to the above induction from $A_i$ to $B_i$.)