I'm working my way through these notes on stochastic calculus:
The following is taken from section 2.20:
In discrete probability, equivalence classes are measurable. (Proof: for any $\omega^\prime$ not equivalent to $\omega$ in $\mathcal{F}$, there is at least one $B_{\omega^\prime} \in \mathcal{F}$ with $\omega \in B_{\omega^\prime}$ but $\omega^\prime \notin B_{\omega^\prime}$. Since there are (at most) countably many $\omega^\prime$, and $\mathcal{F}$ is a $\sigma$-algebra, $A_\omega = \bigcap_{\omega^\prime} B_{\omega^\prime} \in \mathcal{F}$. This $A_\omega$ contains every $\omega_1$ that is equivalent to $\omega$ (why?) and only those.)
Can anyone help me understand this proof?
[ed. note: further context on this question can be found in the notes linked above, or in the answers below.]