I solved an exercise in Billingsley's book Probability and measure. I have no one to correct my solutions and I am rather unsure with this one. So it would be nice if someone could read my attempt and comment, whether it sounds reasonable or not.
Exercise 2.12: A $\sigma$-field (i.e. $\sigma$-algebra) cannot be countably infinite. Its cardinality must be finite or else at least that of the continuum.
Proof: Let $\mathscr F$ be an infinite $\sigma$-field. First I will construct a sequence of pairwise disjoint sets in $\mathscr F$. Take any set $A\in \mathscr F$, $A\neq \emptyset$ and $A\neq \Omega$. The classes $\{A\cap B\,|\, B\in \mathscr F\}$ and $\{A^C\cap B\, |\, B\in \mathscr F\}$ can not be both finite, in this case $\mathscr F$ would be finite. If $\{A^C \cap B\, |\, B\in \mathscr F\}$ is infinite, set $A_1 = A$. Else set $A_1=A^C$. Set $\mathscr F_1 := \{B \cap A_1^C\, |\, B \in \mathscr F\}$. Note that $\mathscr F_1$ is a $\sigma$-field over the subspace $A_1^C$. $\mathscr F_1$ is infinite and $\mathscr F_1 \subset \mathscr F$. Continue and choose $A_2$ from $\mathscr F_1$ in exact the same way as $A_1$ was chosen from $\mathscr F$. This yields a sequence $\{A_1,A_2,...\}$ of pairwise disjoint and non-empty sets in $\mathscr F$.
With every $\omega \in \{0,1\}^{\mathbb N}$ associate the following set $A_\omega$ in $\mathscr F$. \begin{align*} A_\omega := \biguplus_{\omega(i) = 1} A_i \end{align*} If $\omega \neq \omega'$, then $A_\omega$ and $A_{\omega'}$ will be distinct, since the sets in $\{A_1,A_2,...\}$ are pairwise disjoint. Hence, $\mathscr F$ has at least the cardinality of $\{0,1\}^{\mathbb N}$, which is the cardinality of the continuum. $\square$
Thanks a lot in advance. I am especially concerned with the sequence choosing part.